Prime numbers and prime factorization - A complete course in arithmetic It means that the larger number can be composed -- made up -- of the smaller number. 2 is a proper divisor of 10 because 10 can be additively composed of 2's: 5 and 1 are also proper divisors of 10, because 10 can be composed of 5's and 1's. As for 10 itself, we are not interested that 10 is equal to one 10; we want to know what other numbers will compose it. Nevertheless, 10 is called a divisor of itself, but not a proper divisor. The proper divisors do not include the number itself. It is important to note that 1 is a proper divisor of every natural number (except itself), because every natural number is a sum of 1's. That is what a natural number is. 5 = 1 + 1 + 1 + 1 + 1. Problem 1. a) Which numbers are the divisors of 12? To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" ("Reload"). Do the problem yourself first!

1, 2, 3, 4, 6, 12. a) Which are the proper divisors of 12? 1, 2, 3, 4, 6. b) 16 can be composed of which other numbers? 1, 2, 4, 8. Those are the proper divisors of 16. c) 17 can be composed of which other numbers. 1. 1 is its only proper divisor. d) Write all the proper divisors of 29. 1. A prime number is a number whose only proper divisor is 1. Thus a prime number can be composed only of 1's. 17 and 29 are prime numbers. Again, 1 is a proper divisor of every number (except itself), but a prime number has 1 as its only proper divisor. Problem 2. Is 1 itself a prime number? No. 1 has no proper divisors. 1 cannot be composed of other numbers. 1 in this case is the measure. It cannot be measured. Problem 3. Write the first ten prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29. With the exception of 2, then, -- which is the only even prime -- a prime number is a kind of odd number. 3. A composite number. 6 is a composite number, because it can be composed of other numbers besides 1; namely 2 and 3. 1 itself is neither prime nor composite. It is the measure by which we decide whether a number is prime or composite. Square roots When we multiply a number by itself, we say that we have "squared" the number. Thus the square of 5 is 5 × 5 = 25. We then say that 5 is the square root of 25. The square numbers are the numbers we get by squaring a number: 1, 4, 9, 16, 25, and so on. 50, for example, is not a square number, therefore it does not have an exact square root. Its square root is closest to 7, however, because 7 × 7 = 49. Problem 4.

The square root of 175 is closest to what number? 13. 13 × 13= 169. Divisors We can always find the divisors of a number in pairs. One member of the pair will be less than the square root of the number, and the other will be more. (If the number is a square number, then its square root will be its own partner.) For example, here are the pairs of divisors of 24: 1 and 24. (Because 1 × 24 = 24.) 2 and 12. (Because 2 × 12 = 24.) 3 and 8. (Because 3 × 8 = 24.) 4 and 6. (Because 4 × 6 = 24.) Each number on the left is less than the square root of 24, and each number on the right is more. The point is: When we look for divisors of a number, it is necessary to look only up to its square root. Because for each divisor we find less than the square root, we will have found its partner, which is more. Problem 5.

If we are looking for the divisors of 157, up to what number must we look? 12. Because 13 is more than the square root of 157. In this Lesson we will be looking for the prime divisors of a number, and to that we now turn. Prime divisors Every number except 1 is either prime or composite. And if a number is composite, it can be composed of at least one prime number. It will have at least one prime divisor.. (Euclid, VII. 31) 28 is composite. It is composed of fourteen 2's, and it is composed of four 7's. 2 and 7 are prime divisors of 28. Example 1. 63 could be composed of which prime numbers? Answer. Here again are the first few prime numbers: 2, 3, 5, 7, 11, 13, 17. We must test whether 2 is a divisor, or 3, or 5, and so on. But we need test only up to 7, because 11 is more than the square root of 63.

Is 2 a divisor of 63? No, it is not. How do we know? Because 63 is not an even number. And how do we know that? Because even numbers end in 0, 2, 4, 6, or 8. Next, is 3 a divisor of 63? There is a test for divisibility by 3, and it is as follows: A number is divisible by 3 if the sum of the digits is divisible by 3. The sum of the digits of 63 is 6 + 3 = 9, which is divisible by 3. That tells us that 63 can be composed of 3's. Next, is 63 divisible by 5? There is a simple test for divisibility by 5, namely, the number ends in either 0 or 5. 63 does not end in 0 or 5. Therefore, 63 is not divisible by 5. Finally, is 63 divisible by 7? Yes, it is. 63 is equal to nine 7's. 63, then, has could be composed of the prime numbers 3 and 7.

Example 2. Which prime numbers are divisors of 78? Answer. 2 is a prime divisor: 78 = 2 × 39. Now, 39 is composite. Therefore it has a prime divisor: 39 = 3 × 13. 39 is thus divisible by both 3 and 13. And since 78 is divisible by 39, then 78 is also divisible by 3 and 13.. That is called the transitive property of " is divisible by." If a is divisible by b, and b is divisible by c, then a is divisible by c. 78 therefore has three prime divisors: 2, 3, and 13. Problem 6. What is the smallest prime that is a divisor of the following? a) 231. 3. The sum of the digits is divisible by 3. b) 3,165. 3. When numbers are multiplied, they are called factors. Since 30 = 2 × 15, we say that 2 and 15 are factors of 30.

Now, 2 is a prime factor -- but 15 is not. However, 15 = 3 × 5. Therefore we can express 30 as a product of prime factors only: 30 = 2 × 3 × 5. "2 × 3 × 5" is called the prime factorization of 30. And it is unique. That is, apart from the order of the factors: Every composite number can be uniquely factored as a product of prime numbers only. Note: We could have found those same factors by factoring 30 in any way. For example, 30 = 5 × 6. And since 6 = 2 × 3, 30 = 5 × 2 × 3. Apart from the order, we have found the same prime factors. Example 3. Find the prime factorization of 102. Solution. 2 is obviously a prime factor. Its partner will be half of 102 -- 51 ( Lesson 16 ): 102 = 2 × 51.

Because the sum of the digits is divisible by 3, we know that 51 has a prime factor 3. Now, 3 times what is equal to 51? On mentally decomposing 51 into 30 + 21, 51 Problem 8. Find the square factors of each number by writing its prime factorization. a) 112 = 2 × 2 × 2 × 2 × 7 = 16 × 7. b) 450 = 3 × 3 × 5 × 5 × 2 = 3 × 5 × 3 × 5 = 225 × 2. c) 153 = 3 × 51 = 3 × 3 × 17 = 9 × 17. d) 294 = 2 × 147 = 2 × 3 × 49 = 49 × 6. e) 1225 = 25 × 49. 1225 is itself a square number. It is the square of 5 × 7 = 35. Is there a last prime?

As the numbers get larger, the greater the possibility that they will have a divisor, so that there might in fact be a last prime. Now, if there were a last prime, then we could imagine a list that contains every prime up to and including the last one. We will now prove that there is no such list, which is to say, There is no last prime. Here is the theorem: Given any list of prime numbers, there will always be a prime number that is not on the list. Let the following, then, be any list of prime numbers: 2, 3, 5, 7, 11, . . . , P. Now construct the number N which will be the product of every prime on that list: N = 2 × 3 × 5 × 7 × 11 × . . . × P. Every prime on the list is thus a divisor of N. Add 1 to N: N + 1 = (2 × 3 × 5 × 7 × 11 × . . . × P) + 1. The first thing to note is that N + 1 is not on the list, because it is greater than every number on the list.

Now, N + 1 is either prime or composite. If it is prime, then we have found a prime that is not on the list, and the theorem is proved. If N + 1 is composite, then it has a prime factor p. But p is not one of the primes on the list For if it were, then p would be a divisor of both N + 1 and N. That would imply that p divides their difference ( Lesson 11 ), namely 1 -- which is absurd. Therefore, if N + 1 is composite, then there is a prime p that is a divisor of N + 1 but not a divisor of N , which is to say, p is not a prime on the list. Therefore, given any list of primes, there will always be a prime that is not on the list. Which is what we wanted to prove. * A modern enunciation of this theorem is: The number of primes is infinite.

Euclid thus teaches us what we mean, or rather what we should mean, when we say that a collection of numbers is "infinite." It means: "No list of them is complete; there will always be more." That meaning of "infinite" refers to something that we could actually sense -- actually be aware of -- namely a list. It does not refer to something that we cannot be aware of: a list that has no end. * A famous problem in mathematics is the twin prime conjecture. It states that there are infinitely many pairs of primes that differ only by 2. For example, 5 and 7, 17 and 19, 41 and 43. The conjecture has never been proved. What about prime triples, which are three primes that differ only by 2? For example, 3, 5, 7. What do you think? Are there many -- perhaps even an infinite number -- of such triples? Or is 3, 5, 7 the only one? 3, 5, 7 is the only such triple. Because in every sequence of three odd numbers, at least one of them is a multiple of 3.

If the first is a multiple of 3, then the proposition is proved; for example, 21, 23, 25. 21 is not a prime. If the first is 1 more than a multiple of 3, then, on adding 2, the next will be a multiple of 3; for example, 25, 27, 29. Finally, if the first is 1 less than a multiple of 3, then the next will be 1 more, and the third will be a multiple of 3; for example, 35, 37, 39. In the sequence, 3, 5, 7, 3 is the only multiple of 3 that is a prime. Next Lesson: The Largest Known Primes Primes: [ Home || Largest | Proving | How Many? | Mersenne | Glossary | Mailing List ] Note: The correct URL for this page is . The site The Top Twenty is a greatly expanded version of this information. This page summarizes the information on the list of 5000 Largest Known Primes ( updated hourly ). The complete list of is available in several forms . 1.

Introduction An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself. For example, the prime divisors of 10 are 2 and 5; and the first six primes are 2, 3, 5, 7, 11 and 13. ( The first 10,000 , and other lists are available). The Fundamental Theorem of Arithmetic shows that the primes are the building blocks of the positive integers: every positive integer is a product of prime numbers in one and only one way, except for the order of the factors. (This is the key to their importance: the prime factors of an integer determines its properties.) The ancient Greeks proved (ca 300 BC) that there were infinitely many primes and that they were irregularly spaced (there can be arbitrarily large gaps between successive primes ). On the other hand, in the nineteenth century it was shown that the number of primes less than or equal to n approaches n/( log n) (as n gets very large); so a rough estimate for the nth prime is n log n (see the document " How many primes are there?

") The Sieve of Eratosthenes is still the most efficient way of finding all very small primes (e.g., those less than 1,000,000). However, most of the largest primes are found using special cases of Lagrange's Theorem from group theory. See the separate documents on proving primality for more information. In 1984 Samuel Yates defined a titanic prime to be any prime with at least 1,000 digits [ Yates84 , Yates85 ]. When he introduced this term there were only 110 such primes known; now there are over 1000 times that many! And as computers and cryptology continually give new emphasis to search for ever larger primes, this number will continue to grow. Before long we expect to see the first twenty-five million digit prime. If you want to understand a building, how it will react to weather or fire, you first need to know what it is made of. The same is true for the integers--most of their properties can be traced back to what they are made of: their prime factors.

For example, in Euclid's Geometry (over 2,000 years ago), Euclid studied even perfect numbers and traced them back to what we now call Mersenne primes. The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length... Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated. (Carl Friedrich Gauss, Disquisitiones Arithmeticae, 1801) Click here to see all of the Sophie Germain primes on the list of Largest Known Primes. 3. Other Sources of Large Primes Because of the lag time between writing and printing, books can never keep up with the current prime records (that is why this page exists!) However books can provide the mathematical theory behind these records much better than a limited series of web pages can. Recently there have been quite a number of excellent books published on primes and primality proving. Here are some of my favorite: P.

Ribenboim , The new book of prime number records, 3rd edition, Springer-Verlag, New York, 1995. (QA246 .R472). P. Ribenboim , The little book of bigger primes, Springer-Verlag, New York, 2004. (A less mathematical version of the above text.) H. Riesel , Prime numbers and computer methods for factorization, Progress in Mathematics volume 126, Birkäuser Boston, 1994. R. Crandall and C. Pomerance , Prime numbers: a computational perspective, Springer-Verlag, New York, 2001. ISBN 0-387-94777-9. See also [ Bressoud89 ] and [ Cohen93 ] on the page of partially annotated prime references . Also of interest is the Cunningham Project , an effort to factor the numbers in the title of the following book. J.

Brillhart, et al., Factorizations of bn±1 b = 2,3,5,6,7,10,11,12 up to high powers , American Mathematical Society, 1988 [ BLSTW88 ].

The First 1,000 Primes (the 1,000th is 7919) For more information on primes see 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 

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719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997 1009 1013 1019 1021 1031 1033 1039 1049 1051 1061 1063 1069 1087 1091 1093 1097 1103 1109 11

17 1123 1129 1151 1153 1163 1171 1181 1187 1193 1201 1213 1217 1223 1229 1231 1237 1249 1259 1277 1279 1283 1289 1291 1297 1301 1303 1307 1319 1321 1327 1361 1367 1373 1381 1399 1409 1423 1427 1429 1433 1439 1447 1451 1453 1459 1471 1481 1483 1487 1489 149

3 1499 1511 1523 1531 1543 1549 1553 1559 1567 1571 1579 1583 1597 1601 1607 1609 1613 1619 1621 1627 1637 1657 1663 1667 1669 1693 1697 1699 1709 1721 1723 1733 1741 1747 1753 1759 1777 1783 1787 1789 1801 1811 1823 1831 1847 1861 1867 1871 1873 1877 1879

 1889 1901 1907 1913 1931 1933 1949 1951 1973 1979 1987 1993 1997 1999 2003 2011 2017 2027 2029 2039 2053 2063 2069 2081 2083 2087 2089 2099 2111 2113 2129 2131 2137 2141 2143 2153 2161 2179 2203 2207 2213 2221 2237 2239 2243 2251 2267 2269 2273 2281 2287 

2293 2297 2309 2311 2333 2339 2341 2347 2351 2357 2371 2377 2381 2383 2389 2393 2399 2411 2417 2423 2437 2441 2447 2459 2467 2473 2477 2503 2521 2531 2539 2543 2549 2551 2557 2579 2591 2593 2609 2617 2621 2633 2647 2657 2659 2663 2671 2677 2683 2687 2689 2

693 2699 2707 2711 2713 2719 2729 2731 2741 2749 2753 2767 2777 2789 2791 2797 2801 2803 2819 2833 2837 2843 2851 2857 2861 2879 2887 2897 2903 2909 2917 2927 2939 2953 2957 2963 2969 2971 2999 3001 3011 3019 3023 3037 3041 3049 3061 3067 3079 3083 3089 31

09 3119 3121 3137 3163 3167 3169 3181 3187 3191 3203 3209 3217 3221 3229 3251 3253 3257 3259 3271 3299 3301 3307 3313 3319 3323 3329 3331 3343 3347 3359 3361 3371 3373 3389 3391 3407 3413 3433 3449 3457 3461 3463 3467 3469 3491 3499 3511 3517 3527 3529 353

3 3539 3541 3547 3557 3559 3571 3581 3583 3593 3607 3613 3617 3623 3631 3637 3643 3659 3671 3673 3677 3691 3697 3701 3709 3719 3727 3733 3739 3761 3767 3769 3779 3793 3797 3803 3821 3823 3833 3847 3851 3853 3863 3877 3881 3889 3907 3911 3917 3919 3923 3929

 3931 3943 3947 3967 3989 4001 4003 4007 4013 4019 4021 4027 4049 4051 4057 4073 4079 4091 4093 4099 4111 4127 4129 4133 4139 4153 4157 4159 4177 4201 4211 4217 4219 4229 4231 4241 4243 4253 4259 4261 4271 4273 4283 4289 4297 4327 4337 4339 4349 4357 4363 

4373 4391 4397 4409 4421 4423 4441 4447 4451 4457 4463 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 4583 4591 4597 4603 4621 4637 4639 4643 4649 4651 4657 4663 4673 4679 4691 4703 4721 4723 4729 4733 4751 4759 4783 4787 4789 4793 4799 4801 4

813 4817 4831 4861 4871 4877 4889 4903 4909 4919 4931 4933 4937 4943 4951 4957 4967 4969 4973 4987 4993 4999 5003 5009 5011 5021 5023 5039 5051 5059 5077 5081 5087 5099 5101 5107 5113 5119 5147 5153 5167 5171 5179 5189 5197 5209 5227 5231 5233 5237 5261 52

73 5279 5281 5297 5303 5309 5323 5333 5347 5351 5381 5387 5393 5399 5407 5413 5417 5419 5431 5437 5441 5443 5449 5471 5477 5479 5483 5501 5503 5507 5519 5521 5527 5531 5557 5563 5569 5573 5581 5591 5623 5639 5641 5647 5651 5653 5657 5659 5669 5683 5689 569

3 5701 5711 5717 5737 5741 5743 5749 5779 5783 5791 5801 5807 5813 5821 5827 5839 5843 5849 5851 5857 5861 5867 5869 5879 5881 5897 5903 5923 5927 5939 5953 5981 5987 6007 6011 6029 6037 6043 6047 6053 6067 6073 6079 6089 6091 6101 6113 6121 6131 6133 6143

 6151 6163 6173 6197 6199 6203 6211 6217 6221 6229 6247 6257 6263 6269 6271 6277 6287 6299 6301 6311 6317 6323 6329 6337 6343 6353 6359 6361 6367 6373 6379 6389 6397 6421 6427 6449 6451 6469 6473 6481 6491 6521 6529 6547 6551 6553 6563 6569 6571 6577 6581 

6599 6607 6619 6637 6653 6659 6661 6673 6679 6689 6691 6701 6703 6709 6719 6733 6737 6761 6763 6779 6781 6791 6793 6803 6823 6827 6829 6833 6841 6857 6863 6869 6871 6883 6899 6907 6911 6917 6947 6949 6959 6961 6967 6971 6977 6983 6991 6997 7001 7013 7019 7

027 7039 7043 7057 7069 7079 7103 7109 7121 7127 7129 7151 7159 7177 7187 7193 7207 7211 7213 7219 7229 7237 7243 7247 7253 7283 7297 7307 7309 7321 7331 7333 7349 7351 7369 7393 7411 7417 7433 7451 7457 7459 7477 7481 7487 7489 7499 7507 7517 7523 7529 75

37 7541 7547 7549 7559 7561 7573 7577 7583 7589 7591 7603 7607 7621 7639 7643 7649 7669 7673 7681 7687 7691 7699 7703 7717 7723 7727 7741 7753 7757 7759 7789 7793 7817 7823 7829 7841 7853 7867 7873 7877 7879 7883 7901 7907 7919 end.

Prime numbers (2,3,5,7,11,13,...) Prime Numbers Is 2 a prime number? What is prime number? Prime number is a positive natural number that has only two positive natural number divisors - one and itself. The opposite of prime numbers are composite numbers. A composite number is a positive nutural number that has at least one positive divisor other than one or itself. The number 1 is not a prime number by definition - it has only one divisor. The number 0 is not a prime number - it is not a positive number and has infinite number of divisors. The number 15 has divisors of 1,3,5,15 because: 15/1=15 So 15 is not a prime number. The number 13 has only two divisors of 1,13. 13/1=13 So 13 is a prime number.

Prime numbers list List of prime numbers up to 100: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,... Is 0 a prime number? The number 0 is not a prime number. Zero is not a positive number and has infinite number of divisors. Is 1 a prime number? The number 1 is not a prime number by definition. One is has one divisor - itself. Is 2 a prime number? The number 2 is a prime number. Two has 2 natural number divisors - 1 and 2: 2 / 1 = 2

Question: What are the first five prime numbers, in order?
Answer: 2, 3, 5, 7, 11