Question: What is the name of one algebraic generalization prime numbers have inspired?

Answer: prime ideals

Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. <hl> Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals . <hl>

Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and <hl> prime ideals <hl>.

Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and <hl> prime ideals <hl> .

question: What is the name of one algebraic generalization prime numbers have inspired?, context: Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.


Question: Any even number larger than what cannot be considered prime?

Answer: 2

Hence, 6 is not prime. <hl> The image at the right illustrates that 1 2 is not prime: 12 = 3 · 4. <hl> No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

Hence, 6 is not prime. The image at the right illustrates that 1<hl> 2 <hl> is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

The image at the right illustrates that 1 <hl> 2 <hl> is not prime: 12 = 3 · 4.

question: Any even number larger than what cannot be considered prime?, context: Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.


Question: What are the specific divisors of all even numbers larger than 2?

Answer: 1, 2, and n

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. <hl> No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n . <hl> This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely <hl> 1, 2, and n <hl>. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely <hl> 1, 2, and n <hl> .

question: What are the specific divisors of all even numbers larger than 2?, context: Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.


Question: What name is given to any prime number larger than 2?

Answer: odd prime

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. <hl> Accordingly, the term odd prime refers to any prime number greater than 2. <hl> Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term <hl> odd prime <hl> refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

question: What name is given to any prime number larger than 2?, context: Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.


Question: Besides 1,3 and 7, what other number must all primes greater than 5 end with?

Answer: 9

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. <hl> Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9 , since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. <hl>

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or <hl> 9 <hl>, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or <hl> 9 <hl> , since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

question: Besides 1,3 and 7, what other number must all primes greater than 5 end with?, context: Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.


Question: What type of numbers are always multiples of 2?

Answer: even numbers

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. <hl> Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. <hl>

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since <hl> even numbers <hl> are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

question: What type of numbers are always multiples of 2?, context: Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.


Question: What number did early Greeks not regard as a true number?

Answer: 1

<hl> Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. <hl> By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

Most early Greeks did not even consider <hl> 1 <hl> to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

question: What number did early Greeks not regard as a true number?, context: Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".


Question: Who included 1 as the first prime number in the mid 18th century?

Answer: Christian Goldbach

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. <hl> In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. <hl> In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century <hl> Christian Goldbach <hl> listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

question: Who included 1 as the first prime number in the mid 18th century?, context: Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".


Question: In the mid 18th century, who did not concur that 1 should be the first prime number?

Answer: Leonhard Euler

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. <hl> In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. <hl> In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with <hl> Leonhard Euler <hl> -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

question: In the mid 18th century, who did not concur that 1 should be the first prime number?, context: Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".


Question: How many primes were included in Derrick Norman Lehmer's list of prime numbers?

Answer: 10,006,721

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. <hl> For example, Derrick Norman Lehmer's list of primes up to 10,006,721 , reprinted as late as 1956, started with 1 as its first prime. <hl> Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to <hl> 10,006,721 <hl>, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".

For example, Derrick Norman Lehmer's list of primes up to <hl> 10,006,721 <hl> , reprinted as late as 1956, started with 1 as its first prime.

question: How many primes were included in Derrick Norman Lehmer's list of prime numbers?, context: Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".


Question: What type of number do modern mathematicians consider 1 to be?

Answer: its own special category as a "unit"

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. <hl> By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit" . <hl>

Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms <hl> its own special category as a "unit" <hl>.

By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms <hl> its own special category as a "unit" <hl> .

question: What type of number do modern mathematicians consider 1 to be?, context: Most early Greeks did not even consider 1 to be a number, so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number. In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree. In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit".


Question: Which theorem would be invalid if the number 1 were considered prime?

Answer: Euclid's fundamental theorem of arithmetic

<hl> A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. <hl> For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

A large body of mathematical work would still be valid when calling 1 a prime, but <hl> Euclid's fundamental theorem of arithmetic <hl> (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

question: Which theorem would be invalid if the number 1 were considered prime?, context: A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.


Question: The sieve of Eratosthenes would not be valid if what were true?

Answer: if 1 were considered a prime

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. <hl> Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime : a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. <hl> Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly <hl> if 1 were considered a prime <hl>: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

Similarly, the sieve of Eratosthenes would not work correctly <hl> if 1 were considered a prime <hl> : a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1.

question: The sieve of Eratosthenes would not be valid if what were true?, context: A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.


Question: What is another function that primes have that the number 1 does not?

Answer: Euler's totient function

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. <hl> Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function. <hl>

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of <hl> Euler's totient function <hl> or the sum of divisors function.

question: What is another function that primes have that the number 1 does not?, context: A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.


Question: What is one function that prime numbers have that 1 does not?

Answer: the sum of divisors function

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. <hl> Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function . <hl>

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or <hl> the sum of divisors function <hl>.

Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or <hl> the sum of divisors function <hl> .

question: What is one function that prime numbers have that 1 does not?, context: A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.


Question: If 1 were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers?

Answer: only the single number 1

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. <hl> Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1 . <hl> Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output <hl> only the single number 1 <hl>. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.

Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output <hl> only the single number 1 <hl> .

question: If 1 were to be considered as prime what would the sieve of Eratosthenes yield for all other numbers?, context: A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.


Question: What is the name of the Egyptian papyrus that suggests that they may have had knowledge of prime numbers?

Answer: the Rhind papyrus

<hl> There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus , for instance, have quite different forms for primes and for composites. <hl> However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in <hl> the Rhind papyrus <hl>, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in <hl> the Rhind papyrus <hl> , for instance, have quite different forms for primes and for composites.

question: What is the name of the Egyptian papyrus that suggests that they may have had knowledge of prime numbers?, context: There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.


Question: What civilization was the first known to clearly study prime numbers?

Answer: the Ancient Greeks

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. <hl> However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks . <hl> Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from <hl> the Ancient Greeks <hl>. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

However, the earliest surviving records of the explicit study of prime numbers come from <hl> the Ancient Greeks <hl> .

question: What civilization was the first known to clearly study prime numbers?, context: There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.


Question: What work from around 300 BC has significant theorems about prime numbers?

Answer: Euclid's Elements

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. <hl> Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. <hl> Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. <hl> Euclid's Elements <hl> (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

question: What work from around 300 BC has significant theorems about prime numbers?, context: There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.


Question: Who demonstrated how to create a perfect number from a Mersenne prime?

Answer: Euclid

Euclid 's

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. <hl> Euclid 's <hl> Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. <hl> Euclid <hl>'s Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

question: Who demonstrated how to create a perfect number from a Mersenne prime?, context: There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.


Question: What does the Sieve of Eratosthenes do?

Answer: compute primes

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. <hl> The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes , although the large primes found today with computers are not generated this way. <hl>

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to <hl> compute primes <hl>, although the large primes found today with computers are not generated this way.

The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to <hl> compute primes <hl> , although the large primes found today with computers are not generated this way.

question: What does the Sieve of Eratosthenes do?, context: There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.


Question: In what year did Pierre de Fermat declare Fermat's little theorem?

Answer: In 1640

After the Greeks, little happened with the study of prime numbers until the 17th century. <hl> In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). <hl> Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

After the Greeks, little happened with the study of prime numbers until the 17th century. <hl> In 1640 <hl> Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

question: In what year did Pierre de Fermat declare Fermat's little theorem?, context: After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.


Question: Besides Leibniz, what other mathematician proved the validity of Fermat's little theorem?

Answer: Euler

After the Greeks, little happened with the study of prime numbers until the 17th century. <hl> In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler ). <hl> Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and <hl> Euler <hl>). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and <hl> Euler <hl> ).

question: Besides Leibniz, what other mathematician proved the validity of Fermat's little theorem?, context: After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.


Question: Of what form do Fermat numbers take?

Answer: 22n + 1

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). <hl> Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). <hl> However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form <hl> 22n + 1 <hl> are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

question: Of what form do Fermat numbers take?, context: After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.


Question: Of what form do Mersenne primes take?

Answer: 2p − 1

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. <hl> The French monk Marin Mersenne looked at primes of the form 2p − 1 , with p a prime. <hl> They are called Mersenne primes in his honor.

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form <hl> 2p − 1 <hl>, with p a prime. They are called Mersenne primes in his honor.

The French monk Marin Mersenne looked at primes of the form <hl> 2p − 1 <hl> , with p a prime.

question: Of what form do Mersenne primes take?, context: After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.


Question: To what extent did Fermat confirm the validity of Fermat numbers?

Answer: up to n = 4 (or 216 + 1)

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). <hl> Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1) . <hl> However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this <hl> up to n = 4 (or 216 + 1) <hl>. However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this <hl> up to n = 4 (or 216 + 1) <hl> .

question: To what extent did Fermat confirm the validity of Fermat numbers?, context: After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.


Question: What is the most elemental way to test the primality of any integer n?

Answer: trial division

<hl> The most basic method of checking the primality of a given integer n is called trial division . <hl> This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

The most basic method of checking the primality of a given integer n is called <hl> trial division <hl>. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

The most basic method of checking the primality of a given integer n is called <hl> trial division <hl> .

question: What is the most elemental way to test the primality of any integer n?, context: The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Question: What makes the method of trial division more efficient?

Answer: if a complete list of primes up to is known

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. <hl> This routine can be implemented more efficiently if a complete list of primes up to is known —then trial divisions need to be checked only for those m that are prime. <hl> For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently <hl> if a complete list of primes up to is known <hl>—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

This routine can be implemented more efficiently <hl> if a complete list of primes up to is known <hl> —then trial divisions need to be checked only for those m that are prime.

question: What makes the method of trial division more efficient?, context: The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Question: Trial division involves dividing n by every integer m greater than what?

Answer: greater than 1

The most basic method of checking the primality of a given integer n is called trial division. <hl> This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. <hl> If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is <hl> greater than 1 <hl> and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

question: Trial division involves dividing n by every integer m greater than what?, context: The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Question: How many divisions are required to verify the primality of the number 37?

Answer: only three divisions

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. <hl> For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite. <hl>

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, <hl> only three divisions <hl> are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

question: How many divisions are required to verify the primality of the number 37?, context: The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Question: What must the integer m be less than or equal to when performing trial division?

Answer: less than or equal to the square root of n

The most basic method of checking the primality of a given integer n is called trial division. <hl> This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n . <hl> If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and <hl> less than or equal to the square root of n <hl>. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

This routine consists of dividing n by each integer m that is greater than 1 and <hl> less than or equal to the square root of n <hl> .

question: What must the integer m be less than or equal to when performing trial division?, context: The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most . For example, for , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.


Question: How many modern types of primality tests for general numbers n are there?

Answer: two main classes

<hl> Modern primality tests for general numbers n can be divided into two main classes , probabilistic (or "Monte Carlo") and deterministic algorithms. <hl> Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into <hl> two main classes <hl>, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into <hl> two main classes <hl> , probabilistic (or "Monte Carlo") and deterministic algorithms.

question: How many modern types of primality tests for general numbers n are there?, context: Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.


Question: What is the name of one type of modern primality test?

Answer: probabilistic (or "Monte Carlo")

<hl> Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. <hl> Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into two main classes, <hl> probabilistic (or "Monte Carlo") <hl> and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

question: What is the name of one type of modern primality test?, context: Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.


Question: What is the name of another type of modern primality test?

Answer: deterministic

<hl> Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. <hl> Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and <hl> deterministic <hl> algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

question: What is the name of another type of modern primality test?, context: Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.


Question: What type of algorithm is trial division?

Answer: deterministic

<hl> Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. <hl> Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and <hl> deterministic <hl> algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

question: What type of algorithm is trial division?, context: Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.


Question: When using a probabilistic algorithm, how is the probability that the number is composite expressed mathematically?

Answer: 1/(1-p)n

Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. <hl> If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n , which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. <hl> On the other hand, if the test ever fails, then we know that the number is composite.

Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is <hl> 1/(1-p)n <hl>, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

If we repeat the test n times and pass every time, then the probability that our number is composite is <hl> 1/(1-p)n <hl> , which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime.

question: When using a probabilistic algorithm, how is the probability that the number is composite expressed mathematically?, context: Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.


Question: What is one straightforward case of a probabilistic test?

Answer: the Fermat primality test,

<hl> A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. <hl> If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number.

A particularly simple example of a probabilistic test is <hl> the Fermat primality test, <hl> which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number.

question: What is one straightforward case of a probabilistic test?, context: A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number.