aarjaneiro/planetmath huggingface dataset:
prime signature

prime signature

The

prime signature

of an

integer

n

is the list of nonzero exponents

a

i

from the

integer factorization

n

=

∏

i

=

1

∞

p

i

a

i

,

(with

p

i

being the

i

th prime) sorted in

ascending order

(

http://planetmath.org/AscendingOrder

) but with duplicates retained. Three examples: the prime signature of 10368 is (4, 7), the prime signature of 10369 is (1), the prime signature of 10370 is (1, 1, 1, 1).

The prime signature of a number is insufficient to uniquely identify it. Numbers like 34992 and 514714375 also have prime signatures of (4, 7). However, prime signatures can identify some kinds of numbers: the primes have

signature

(1); the squares of primes have signature (2), while other

semiprimes

have signature (1, 1);

sphenic numbers

have signature (1, 1, 1); etc. But while other kinds of numbers have different signatures among their members, some

generalizations

can still be made, such as that

highly composite numbers

have prime signatures in reverse order of the factorization as usually stated with the primes from 2 up; or that

Achilles numbers

don’t have any 1s in their prime signature but the

greatest common divisor

of the numbers in the prime signature is 1.

Title

prime signature

Canonical name

PrimeSignature

Date of creation

2013-03-22 18:51:50

Last modified on

2013-03-22 18:51:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:27 2018 by

LaTeXML
primorial

primorial

The

primorial

of

n

, or

n

⁢

#

, is the

product

of the first

n

consecutive primes, thus:

∏

i

=

1

n

p

i

(

p

i

is the

i

th

prime number

).

The first few primorials are 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130; these are listed in A002110 of Sloane’s OEIS. Sometimes the notation

n

⁢

#

is used to refer to the product of all primes

p

<

π

⁢

(

n

)

, where

π

is the

prime counting function

(so then

4

⁢

#

=

6

rather than 210).

Primorials are used in the classic proof that there are infinitely many primes: assuming that there are exactly

n

primes and no more,

n

⁢

#

+

1

is a number that is not

divisible

by any of the existing primes, but if that is a prime then it contradicts the initial

assumption

.

If, in reckoning the

sieve of Eratosthenes

, one strikes out again numbers that have already been struck off, the

sequence

of the smallest number struck off

n

times is precisely the sequence of the primorials.

Any

highly composite number

(with the exception of 1) can be expressed as a product of primorials in at least one way.

Title

primorial

Canonical name

Primorial

Date of creation

2013-03-22 16:00:23

Last modified on

2013-03-22 16:00:23

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:29 2018 by

LaTeXML
primorial prime

primorial prime

A

primorial prime

is a number that is one less or one more than a

primorial

and is also a

prime number

. The first few primorial primes are: 5, 7, 29, 31, 211, 2309, 2311, 30029.

Title

primorial prime

Canonical

name

PrimorialPrime

Date of creation

2013-03-22 16:19:22

Last modified on

2013-03-22 16:19:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:30 2018 by

LaTeXML
probability that two positive integers are relatively prime

probability that two positive integers are relatively prime

The

probability

that two

positive integers

chosen randomly are
relatively prime is

6

π

2

=

0.60792710185

⁢

…

.

At first glance this “naked” result is beautiful, but no
suitable definition is there: there isn’t a

probability space

defined. Indeed, the word “probability” here is an abuse of
language.
So, now, let’s write the mathematical statement.

For each

n

∈

ℤ

+

, let

S

n

be the set

{

1

,

2

,

…

,

n

}

×

{

1

,

2

,

…

,

n

}

and define

Σ

n

to be the powerset of

S

n

. Define

μ

:

Σ

n

→

ℝ

by

μ

⁢

(

E

)

=

|

E

|

/

|

S

n

|

. This makes

(

S

n

,

Σ

n

,

μ

)

into a probability space.

We wish to consider the event of some

(

x

,

y

)

∈

S

n

also being
in the set

A

n

=

{

(

a

,

b

)

∈

S

n

:

gcd

⁡

(

a

,

b

)

=

1

}

.
The probability of this event is

P

(

(

x

,

y

)

∈

A

n

)

=

∫

S

n

χ

A

n

d

μ

=

|

A

n

|

|

S

n

|

.

Our statement is thus the following. For each

n

∈

ℤ

+

,
select random

integers

x

n

and

y

n

with

1

≤

x

n

,

y

n

≤

n

.
Then the limit

lim

n

→

∞

P

(

(

x

n

,

y

n

)

∈

A

n

)

exists and

lim

n

→

∞

P

(

(

x

n

,

y

n

)

∈

A

n

)

=

6

π

2

.

In other words, as

n

gets large, the

fraction

of

|

S

n

|

consisting of relatively prime pairs of positive integers tends to

6

/

π

2

.

References

1

Challenging Mathematical Problems with Elementary Solutions, A.M. Yaglom and I.M. Yaglom, Vol. 1, Holden-Day, 1964. (See Problems 92 and 93)

Title

probability that two positive integers are relatively prime

Canonical name

ProbabilityThatTwoPositiveIntegersAreRelativelyPrime

Date of creation

2013-03-22 14:56:08

Last modified on

2013-03-22 14:56:08

Owner

mps (409)

Last modified by

mps (409)

Numerical id

21

Author

mps (409)

Entry type

Result

Classification

msc 11A41

Classification

msc 11A05

Classification

msc 11A51

Generated on Fri Feb 9 15:17:32 2018 by

LaTeXML
probable prime

probable prime

A sufficiently large

odd integer

q

believed to be a

prime number

because it has passed some preliminary

primality test

relative to a given base, or a pattern suggests it might be prime, but it has not yet been subjected to a conclusive primality test.

For primes with no specific form, it is required to test every potential

prime factor

p

<

q

to be absolutely sure that

q

is in fact a prime. For Mersenne

probable primes

, the

Lucas-Lehmer test

is accepted as a conclusive primality test.

Once a probable prime is conclusively shown to be a prime, it of course loses the label ”probable.” It also loses it if conclusively shown to be

composite

, but in that case it might then be called a

pseudoprime

(

http://planetmath.org/PseudoprimeP

) relative to base

a

.

Title

probable prime

Canonical name

ProbablePrime

Date of creation

2013-03-22 15:53:46

Last modified on

2013-03-22 15:53:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:34 2018 by

LaTeXML
proof of infinitude of primes

proof of infinitude of primes

We begin by noting a fact about factorizations. Suppose that

n

>

0

is an

integer

which has a

prime factorization

n

=

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

m

k

m

.

Then, because

2

is the smallest

prime number

, we must have

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

m

k

m

≥

2

k

1

⁢

2

k

2

⁢

⋯

⁢

2

k

m

=

2

k

1

+

k

2

+

⋯

+

k

m

,

so

n

≥

2

k

1

+

k

2

+

⋯

+

k

m

.

Assume that there were only a finite number of prime numbers

p

1

,

p

2

,

…

⁢

p

m

.
By the above-noted fact, given a

positive integer

j

, every integer n such that

2

j

≥

n

>

0

could be
expressed as

n

=

p

1

k

1

⁢

p

2

k

2

⁢

⋯

⁢

p

m

k

m

with

k

1

+

k

2

+

⋯

+

k

m

≤

j

.

This, however, leads to a

contradiction

because it would imply that there exist more
integers than possible factorizations despite the fact that every integer is supposed
to have a prime factorization. To see this, let us over-count the number of
factorizations. A factorization being specified by an

m

-tuplet of integers

k

1

,

k

2

,

…

,

k

n

such that

k

1

+

k

2

+

⋯

+

k

m

≤

j

, the number of
factorizations is equal to the number of such tuplets. Now, for all

i

we must have

0

≤

k

i

≤

j

, so there are not more than

(

j

+

1

)

m

such tuplets available.
However, for all

m

, one can choose

j

such that

2

j

>

j

m

. For such a choice
of

j

we could not make ends meet — there are not enough possible factorizations
available to handle all integers, so we conclude that there must be more than

m

primes
for any integer

m

, i.e. that the number of primes is

infinite

.

To make this exposition self-contained, we conclude with a proof that, for every

m

,
there exists a

j

such that

2

j

>

(

j

+

1

)

m

. We begin by showing that,
for every integer

a

≥

1

, we have

2

a

>

a

. This is an easy

induction

. When

a

=

1

,
we have

2

1

=

2

>

1

. If

2

a

>

a

for some

a

>

1

, then

2

a

+

1

=

2

a

+

2

a

>

a

+

a

>

a

+

1

.
Hence, by induction,

2

a

>

a

for all

a

≥

1

.

From this starting point, we obtain the desired inequality by

algebraic

manipulation.
Suppose that

a

>

2

. Multiplying both sides by

2

, we get

2

a

+

1

>

2

⁢

a

>

a

+

2

.
Setting

a

=

m

-

2

, we have

2

m

-

1

>

m

, or

2

m

>

2

⁢

m

. Raising both sides to the

2

⁢

m

-th power,

2

2

⁢

m

2

>

2

2

⁢

m

⁢

m

2

⁢

m

=

2

m

⁢

(

2

⁢

m

2

)

m

≥

2

⁢

(

2

⁢

m

2

)

m

. Setting

j

=

(

2

⁢

m

2

-

1

)

, this becomes

2

j

>

(

j

+

1

)

2

.

Title

proof of infinitude of primes

Canonical name

ProofOfInfinitudeOfPrimes

Date of creation

2013-12-27 17:11:49

Last modified on

2013-12-27 17:11:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

15

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 11A41

Generated on Fri Feb 9 15:17:36 2018 by

LaTeXML
proof that 4^x exceeds the product of the primes up to x

proof that

4

x

exceeds the product of the primes up to

x

Statement. (Erdős & Surányi) Given the

prime counting function

π

⁢

(

x

)

and notating the

i

th prime as

p

i

, the inequality

4

x

>

∏

i

=

1

π

⁢

(

x

)

p

i

is always true for any nonnegative

x

. In other words, the

x

th power of 4 is greater than the

primorial

π

⁢

(

x

)

⁢

#

.

Note:

⌊

x

⌋

is the

integer

nearest to the real number

x

that’s not greater than

x

, while

⌈

x

⌉

is the nearest integer to

x

that’s not smaller than

x

.

Proof. It takes very little computational effort to verify this is true for

x

set to 1 or 2. For larger

x

we can high-ball the primorial

π

⁢

(

x

)

⁢

#

as

2

⁢

∏

i

=

1

⌈

x

2

⌉

2

⁢

i

+

1

(that is, twice the product of the

odd numbers

from 1 to

x

) and rephrase

4

x

as

2

⁢

∏

i

=

1

2

⁢

x

-

1

2

.

The first factor is obviously the same for both expressions, both iterators have the same start value, but cleary the iterator end values satisfy the inequality

(

2

⁢

x

-

1

)

>

⌈

x

2

⌉

as long as

x

>

2

. The first expression being iteratively multiplied clearly increases, but the second expression being iteratively multiplied is static (being a 2). This line of thought has failed to yield the desired result.

What if instead we divide both expressions by 2 and take their base 2

logarithms

? Obviously, for half

4

x

,

log

2

⁡

(

∏

i

=

1

2

⁢

x

-

1

2

)

=

2

⁢

x

-

1

,

giving us the

sequence

of odd numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, etc. Not quite so obviously, for half our primorial high-ball, the base 2 logarithm is

log

⁡

(

⌈

x

2

⌉

!!

)

l

⁢

o

⁢

g

⁢

(

2

)

,

(where

log

⁡

(

x

)

is the

natural logarithm

and

n

!!

is the

double factorial

), giving us the sequence (to 5 decimal places) 0, 1.58496, 1.58496, 3.90689, 3.90689, 6.71425, 6.71425, 9.88417, 9.88417, 13.3436, 13.3436, 17.044, 17.044, 20.9509, 20.9509, 25.0384, 25.0384, 29.2863, 29.2863, etc.

Unfortunately, both of these strategies are doomed to failure because

eventually

our high-ball for the primorial overtakes

4

x

. What Erdős and Surányi do instead is rephrase

x

as

x

=

2

⁢

n

+

1

and then construct the

binomial coefficient

(

2

⁢

n

+

1

⁢

n

)

and showing the relationship of this to

2

x

.

References

1

Paul Erdős & János Surányi

Topics in the theory of numbers

New York: Springer (2003): 5.11

Title

proof that

4

x

exceeds the product of the primes up to

x

Canonical name

ProofThat4xExceedsTheProductOfThePrimesUpToX

Date of creation

2013-03-22 17:04:48

Last modified on

2013-03-22 17:04:48

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Proof

Classification

msc 11A41

Classification

msc 11A25

Classification

msc 11N05

Generated on Fri Feb 9 15:17:43 2018 by

LaTeXML
proof that n^2-n+41 is prime for 0≤n≤40

proof that

n

2

-

n

+

41

is prime for

0

≤

n

≤

40

We show that for

0

≤

n

≤

40

,

n

2

-

n

+

41

is prime. Of course, this can easily be seen by considering the

41

cases, but the proof given here is illustrative of why the statement is true.

Recall that there is only one

reduced

(

http://planetmath.org/IntegralBinaryQuadraticForms

)

integral binary quadratic form

of

discriminant

-

163

; that form is

x

2

+

x

⁢

y

+

41

⁢

y

2

. The smallest prime that is represented by that form is

41

. For suppose

p

=

x

2

+

x

⁢

y

+

41

⁢

y

2

and

p

<

41

. Then obviously

y

=

0

, so

p

=

x

2

, which is impossible. Since

equivalent

forms represent the same set of integers, it follows that any form of discriminant

-

163

represents no primes less than

41

.

Now suppose

n

2

-

n

+

41

is

composite

for some

n

≤

40

. Then

n

2

-

n

+

41

≤

40

2

-

40

+

41

<

41

2

and thus

n

2

-

n

+

41

has a

prime factor

q

<

41

. Write

n

2

-

n

+

41

=

q

⁢

c

; then

q

⁢

x

2

+

(

2

⁢

n

-

1

)

⁢

x

⁢

y

+

c

⁢

y

2

represents

q

(

x

=

1

,

y

=

0

); its discriminant is

(

2

⁢

n

-

1

)

2

-

4

⁢

q

⁢

c

=

4

⁢

n

2

-

4

⁢

n

+

1

-

4

⁢

(

n

2

-

n

+

41

)

=

-

163

Since there is only one

equivalence class

of forms with discriminant

-

163

,

q

⁢

x

2

+

(

2

⁢

n

-

1

)

⁢

x

⁢

y

+

c

⁢

y

2

is equivalent to

x

2

+

x

⁢

y

+

41

⁢

y

2

and thus represents the same

integers

. But we know that

x

2

+

x

⁢

y

+

41

⁢

y

2

cannot represent any prime

<

41

, so cannot represent

q

.

Contradiction

. So

n

2

-

n

+

41

is prime for

n

≤

40

.

This proof works equally well for the other cases mentioned in the parent article, since for each of those cases, there is only one reduced form (

http://planetmath.org/IntegralBinaryQuadraticForms

) of the appropriate discriminant, which is

1

-

4

⁢

p

.

Title

proof that

n

2

-

n

+

41

is prime for

0

≤

n

≤

40

Canonical name

ProofThatN2n41IsPrimeFor0leqNleq40

Date of creation

2013-03-22 16:55:48

Last modified on

2013-03-22 16:55:48

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Proof

Classification

msc 11A41

Generated on Fri Feb 9 15:17:46 2018 by

LaTeXML
Pythagorean prime

Pythagorean prime

A

Pythagorean prime

p

is a

prime number

of the form

4

⁢

n

+

1

. The first few are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, etc., listed in A002144 of Sloane’s OEIS. Because of its form, a Pythagorean prime is the

sum of two squares

, e.g., 29 = 25 + 4. In fact, with the exception of 2, these are the only primes that can be represented as the sum of two squares (thus, in Waring’s problem, all other primes require three or four squares).

Though Pythagorean primes are primes on the line of real

integers

, they are not

Gaussian primes

in the

complex plane

. Expressing a Pythagorean prime as

a

2

+

b

2

(it doesn’t matter whether

a

<

b

or viceversa) leads to the complex factorization by simple plugging in of the values thus:

p

=

(

a

+

b

⁢

i

)

⁢

(

a

-

b

⁢

i

)

, where

i

is the

imaginary unit

.

Title

Pythagorean prime

Canonical name

PythagoreanPrime

Date of creation

2013-03-22 16:53:53

Last modified on

2013-03-22 16:53:53

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:51 2018 by

LaTeXML
Ramanujan prime

Ramanujan prime

The

n

th

p

is the smallest prime such that there are at least

n

primes between

x

and

2

⁢

x

for any

x

such that

2

⁢

x

>

p

. So, given the

prime counting function

π

⁢

(

x

)

, then for the

n

th

Ramanujan prime

p

it is always the case that

π

⁢

(

2

⁢

x

)

-

π

⁢

(

x

)

≥

n

when

2

⁢

x

>

p

. These primes arise from

Srinivasa Ramanujan

’s proof of Bertrand’s postulate. The first few are 2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, listed in A104272 of Sloane’s OEIS.

For example, the third Ramanujan prime is 17. We can verify that there are three primes between 8.5005 and 17.001 (namely 11, 13, 17), that there are also three primes between 9 and 18 (the same as before), more than three primes between 10 and 20 (namely the

prime quadruplet

11, 13, 17. 19), etc. Furthermore, we can verify that no prime smaller than 17 satisfies this condition by finding a single

counterexample

for the smaller primes, specifically: setting

x

=

7

we have

2

⁢

x

=

14

, which is greater than 2, 3, 5, 7, 11 and 13, and we verify that there are only two primes between 7 and 14 (namely 11 and 13).

References

1

. Ramanujan, “A proof of Bertrand’s postulate”

J. Indian Math. Society

11

, 1919: 181 - 182

2

. Sondow, “Ramanujan primes and Bertrand’s postulate”

Amer. Math. Monthly

116

, 2009: 630 - 635

Title

Ramanujan prime

Canonical

name

RamanujanPrime

Date of creation

2013-03-22 16:38:46

Last modified on

2013-03-22 16:38:46

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:52 2018 by

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safe prime

safe prime

Given a prime

q

, if

q

-

1

2

=

p

, where

p

is another

prime number

, then

q

is a

safe prime

. It follows that

q

-

1

=

2

⁢

p

(and thus a

semiprime

) and

p

is a

Sophie Germain prime

.

Every safe prime matches to a Sophie Germain prime this way, and some safe primes are Sophie Germain primes themselves too,

matching

to yet another safe prime. A set of primes linked in this way is called a

Cunningham chain

of the first kind. For example, 2, 5, 11, 23, 47. With Pocklington’s criterion, one can prove the

primality

of a safe prime if one has first proven the primality of its matching Sophie Germain prime.

The first few safe primes are 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, … These are listed in A005385 of Sloane’s OEIS. As of 2010, the largest known safe prime is

183027

×

2

265441

-

1

.

Safe primes have been used in various methods of

cryptography

, but the safety of their use depends not just on these mathematical properties but also on their being large enough that their

multiples

can’t be factored in a reasonable period of time by contemporary computers.

0.1

External links

http://primes.utm.edu/primes/page.php?id=92223

Information

on the largest known Sophie Germain prime from the Prime Pages.

Title

safe prime

Canonical name

SafePrime

Date of creation

2013-03-22 15:49:31

Last modified on

2013-03-22 15:49:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:17:58 2018 by

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semiprime

semiprime

A

composite number

which is the product of two (possibly equal) primes is called

semiprime

. Such numbers are sometimes also called 2-

almost primes

. For example:

•

1 is not a semiprime because it is not a composite number or a prime,

•

2 is not a semiprime, as it is a prime,

•

4 is a semiprime, since

4

=

2

⋅

2

,

•

8 is not a semiprime, since it is a product of three primes (

8

=

2

⋅

2

⋅

2

),

•

2003 is not a semiprime, as it is a prime,

•

2005 is a semiprime, since

2005

=

5

⋅

401

,

•

2007 is not a semiprime, since it is a product of three primes (

2007

=

3

⋅

3

⋅

223

).

The first few semiprimes are

4

,

6

,

9

,

10

,

14

,

15

,

21

,

22

,

25

,

26

,

33

,

34

,

35

,

38

,

39

,

46

,

49

,

51

,

55

,

57

,

58

,

62

,

…

(

http://www.research.att.com/cgi-bin/access.cgi/as/njas/

sequences

/eisA.cgi?Anum=001358

Sloane’s sequence A001358
). The

Moebius function

μ

⁢

(

n

)

for semiprimes can be only equal to 0 or 1. If we form an

integer sequence

of values of

μ

⁢

(

n

)

for semiprimes we get a binary sequence:

0

,

1

,

0

,

1

,

1

,

1

,

1

,

1

,

0

,

1

,

1

,

1

,

1

,

1

,

1

,

1

,

0

,

1

,

1

,

1

,

1

,

1

,

…

. (

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=072165

Sloane’s sequence A072165
).

All the squares of primes are also semiprimes. The first few squares of primes are then

4

,

9

,

25

,

49

,

121

,

169

,

289

,

361

,

529

,

841

,

961

,

1369

,

1681

,

1849

,

2209

,

2809

,

3481

,

3721

,

4489

,

5041

,

…

. (

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=001248

Sloane’s sequence A001248
). The Moebius function

μ

⁢

(

n

)

for the squares of primes is always equal to 0 as it is equal to 0 for all squares.

Title

semiprime

Canonical name

Semiprime

Date of creation

2013-03-22 12:49:22

Last modified on

2013-03-22 12:49:22

Owner

drini (3)

Last modified by

drini (3)

Numerical id

9

Author

drini (3)

Entry type

Definition

Classification

msc 11A41

Synonym

semi-prime

Synonym

2-almost prime

Defines

almost prime

Generated on Fri Feb 9 15:18:01 2018 by

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Shantha prime

Shantha prime

A

Shantha prime

p

is a

prime number

of the form

p

=

3

q

-

2

with

q

being a Mangammal prime (

http://planetmath.org/MangammalPrime

). The smallest Shantha prime is

7

=

3

2

-

2

. The next is

3

541

-

2

and has 259 digits. Shantha primes are very rare among the smaller numbers. The above formulation generates mostly

composite numbers

.

References

1

A. K. Devaraj, ”Euler’s

Generalization

of Fermat’s Theorem-A Further Generalization”, in

Proceedings of Hawaii International Conference on Statistics, Mathematics & Related Fields

, 2004.

Title

Shantha prime

Canonical

name

ShanthaPrime

Date of creation

2013-03-22 17:49:54

Last modified on

2013-03-22 17:49:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

26

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A41

Generated on Fri Feb 9 15:18:03 2018 by

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sieve of Eratosthenes

sieve of Eratosthenes

The

sieve of Eratosthenes

is a simple algorithm for generating a list of the

prime numbers

between

1

and some

integer

N

≥

2

.

Let

p

=

2

, which is of course known to be prime. Mark all

positive

multiples

of

p

(e.g.

2

,

4

,

6

,

8

⁢

…

), up until

N

, as

composite

. Now let

p

be the smallest number not marked as composite (in this case,

3

); it must be the next prime. Again, mark all positive multiples of

p

as composite. Continue this process while

p

≤

N

. When done, all numbers less than

N

that have not been marked as composite are prime.

For many years, the sieve of Eratosthenes was the fastest known algorithm for listing primes. Today, there are faster methods, such as a

quadratic sieve

.

Title

sieve of Eratosthenes

Canonical name

SieveOfEratosthenes

Date of creation

2013-03-22 12:39:34

Last modified on

2013-03-22 12:39:34

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A41

Related topic

BrunsPureSieve

Generated on Fri Feb 9 15:18:04 2018 by

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