aarjaneiro/planetmath huggingface dataset:
functional equation of the Riemann zeta function

functional equation of the Riemann zeta function

Let

Γ

denote the

gamma function

,

ζ

the

Riemann zeta function

and

s

any

complex number

. Then

π

s

-

1

2

⁢

Γ

⁢

(

1

-

s

2

)

⁢

ζ

⁢

(

1

-

s

)

=

π

-

s

2

⁢

Γ

⁢

(

s

2

)

⁢

ζ

⁢

(

s

)

.

Though the equation appears too intricate to be of any use, the inherent in the formula makes this the simplest method of evaluating

ζ

⁢

(

s

)

at points to the left of the

critical strip

.

Title

functional equation

of the Riemann zeta function

Canonical name

FunctionalEquationOfTheRiemannZetaFunction

Date of creation

2013-03-22 13:54:29

Last modified on

2013-03-22 13:54:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11M99

Related topic

ClassNumberFormula

Generated on Fri Feb 9 15:38:05 2018 by

LaTeXML
value of the Riemann zeta function at s=2

value of the Riemann zeta function at

s

=

2

Here we present an application of Parseval’s equality to

number
theory

. Let

ζ

⁢

(

s

)

denote the

Riemann zeta function

. We will
compute the value

ζ

⁢

(

2

)

with the help of

Fourier analysis

.

Example:

Let

f

:

ℝ

→

ℝ

be the “identity”

function

,
defined by

f

⁢

(

x

)

=

x

,

for all

⁢

x

∈

ℝ

.

The

Fourier series

of this function has been computed in the entry
example of Fourier series.

Thus

f

⁢

(

x

)

=

x

=

a

0

f

+

∑

n

=

1

∞

(

a

n

f

⁢

cos

⁡

(

n

⁢

x

)

+

b

n

f

⁢

sin

⁡

(

n

⁢

x

)

)

=

∑

n

=

1

∞

(

-

1

)

n

+

1

⁢

2

n

⁢

sin

⁡

(

n

⁢

x

)

,

∀

x

∈

(

-

π

,

π

)

.

Parseval’s

theorem

asserts that:

1

π

⁢

∫

-

π

π

f

2

⁢

(

x

)

⁢

𝑑

x

=

2

⁢

(

a

0

f

)

2

+

∑

k

=

1

∞

[

(

a

k

f

)

2

+

(

b

k

f

)

2

]

.

So we apply this to the function

f

⁢

(

x

)

=

x

:

2

⁢

(

a

0

f

)

2

+

∑

k

=

1

∞

[

(

a

k

f

)

2

+

(

b

k

f

)

2

]

=

∑

n

=

1

∞

4

n

2

=

4

⁢

∑

n

=

1

∞

1

n

2

and

1

π

⁢

∫

-

π

π

f

2

⁢

(

x

)

⁢

𝑑

x

=

1

π

⁢

∫

-

π

π

x

2

⁢

𝑑

x

=

2

⁢

π

2

3

.

Hence by Parseval’s equality

4

⁢

∑

n

=

1

∞

1

n

2

=

2

⁢

π

2

3

and hence

ζ

⁢

(

2

)

=

∑

n

=

1

∞

1

n

2

=

π

2

6

.

Title

value of the Riemann zeta function at

s

=

2

Canonical

name

ValueOfTheRiemannZetaFunctionAtS2

Date of creation

2013-03-22 13:57:16

Last modified on

2013-03-22 13:57:16

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

15

Author

alozano (2414)

Entry type

Theorem

Classification

msc 11M99

Classification

msc 42A16

Related topic

ExampleOfFourierSeries

Related topic

PersevalEquality

Related topic

ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers

Related topic

ValueOfRiemannZetaFunctionAtS4

Related topic

ValueOfDirichletEtaFunctionAtS2

Related topic

APathologicalFunctionOfRiemann

Related topic

KummersAccelerationMethod

Generated on Fri Feb 9 15:38:13 2018 by

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values of the Riemann zeta function in terms of Bernoulli numbers

values of the Riemann zeta function in terms of Bernoulli numbers

Theorem

.

Let

k

be an

even integer

and let

B

k

be the

k

th

Bernoulli number

. Let

ζ

⁢

(

s

)

be the

Riemann zeta function

. Then:

ζ

⁢

(

k

)

=

2

k

-

1

⁢

|

B

k

|

⁢

π

k

k

!

Moreover, by using the functional equation (

http://planetmath.org/RiemannZetaFunction

)
, one calculates for all

n

≥

1

:

ζ

⁢

(

1

-

n

)

=

(

-

1

)

n

+

1

⁢

B

n

n

which shows that

ζ

⁢

(

1

-

n

)

=

0

for

n

≥

3

odd. For

k

≥

2

even, one has:

ζ

⁢

(

1

-

k

)

=

-

B

k

k

.

Remark

.

The zeroes of the

zeta function

shown above,

ζ

⁢

(

1

-

n

)

=

0

for

n

≥

3

odd, are usually called the

trivial

zeroes of the Riemann zeta function, while the

non-trivial

zeroes are those in the

critical strip

.

Title

values of the Riemann zeta function in terms of Bernoulli numbers

Canonical name

ValuesOfTheRiemannZetaFunctionInTermsOfBernoulliNumbers

Date of creation

2013-03-22 15:12:07

Last modified on

2013-03-22 15:12:07

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 11M99

Related topic

BernoulliNumber

Related topic

ValueOfTheRiemannZetaFunctionAtS2

Generated on Fri Feb 9 15:38:15 2018 by

LaTeXML
Brun’s constant for prime quadruplets

Brun’s constant for prime quadruplets

Brun’s constant for

prime quadruplets

is the sum of the

reciprocals

of all prime quadruplets

B

4

=

∑

p

p

+

2

⁢

is prime

p

+

6

⁢

is prime

p

+

8

⁢

is prime

(

1

p

+

1

p

+

2

+

1

p

+

6

+

1

p

+

8

)

≈

0.8705883800

.

Viggo Brun proved that the constant exists by using a new sieving method, which later became known as Brun’s sieve (

http://planetmath.org/BrunsPureSieve

).

Title

Brun’s constant for prime quadruplets

Canonical

name

BrunsConstantForPrimeQuadruplets

Date of creation

2013-03-22 16:06:23

Last modified on

2013-03-22 16:06:23

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Classification

msc 11N36

Synonym

Brun’s constant for prime quadruples

Synonym

Brun’s constant for prime quartets

Generated on Fri Feb 9 15:38:21 2018 by

LaTeXML
Carol number

Carol number

Given

n

, compute

4

n

-

2

n

+

1

-

1

or

(

2

n

-

1

)

2

-

2

or

(

2

2

⁢

n

-

1

)

-

2

n

+

1

or

∑

i

≠

n

+

1

i

=

0

2

⁢

n

2

i

.

Any of these formulas gives the

Carol number

for

n

. Regardless of how they’re computed, these numbers are almost

repunits

in binary, needing only the

addition

of

2

n

+

1

to become so.

The first few Carol numbers are 7, 47, 223, 959, 3967, 16127, 65023, 261119, 1046527, 4190207, 16769023 (listed in A093112 of Sloane’s OEIS). Every third Carol number is

divisible

by 7, thus prime Carol numbers can’t have

n

=

3

⁢

x

+

2

(except of course for

n

=

2

. The largest Carol number known to be prime is

(

2

248949

-

1

)

2

-

2

, found by Japke Rosink using MultiSieve and OpenPFGW in March 2006.

Title

Carol number

Canonical

name

CarolNumber

Date of creation

2013-03-22 16:13:10

Last modified on

2013-03-22 16:13:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:22 2018 by

LaTeXML
Chen prime

Chen prime

If for a

prime number

p

it holds that

p

+

2

is either a prime or a

semiprime

, then

p

is called a

Chen prime

. The name was assigned by Ben Green and Terrence Tao in recognition of Chen’s

theorem

that every sufficiently large

even number

can be written as the sum of a prime and a semiprime. To give two examples of Chen primes: 41 is a Chen prime since 43 is also a prime, but 43 is itself not a Chen prime because 45 has one factor too many to be a semiprime; 47 is a Chen prime since 49, the square of a prime, is a semiprime.

Chen Jingrun

proved that there are infinitely many Chen primes, which could turn out to be a step towards proving the

twin prime conjecture

. Just looking at say,

p

<

100

, it would appear that there are more Chen primes than non-Chen primes. (The former are listed in A109611 of Sloane’s OEIS, the latter in A102540). However, counting up to 17107, there are 986 Chen primes and 986 non-Chen; after that, the density of Chen primes gradually thins.

In 2005, Green and Tao proved that there are infinitely many Chen primes in

arithmetic progression

. Jens Kruse Andersen and friends found this example, in which each prime has more than 3000 base 10 digits each:

(

(

3850324118

+

892819689

⁢

n

)

⁢

2411

⁢

#

+

1

)

⁢

(

4787

⁢

#

+

1

)

-

2

where

-

1

<

n

<

3

and

p

⁢

#

is a

primorial

.

Rudolf Ondrejka constructed this

magic square

using only Chen primes:

[

17

89

71

113

59

5

47

29

101

]

The

magic constant

is 177.

References

1

J. Chen, “On the Representation of a Large Even Integer as the Sum of a Prime and the

Product

of at Most Two Primes”

Sci. Sinica

16

, pp. 157 - 176 (1973)

2

B. Green and T. Tao, “

Restriction

theory of the Selberg sieve, with applications”, pp. 5, 14, 18 - 19, 21 (2005)

Title

Chen prime

Canonical

name

ChenPrime

Date of creation

2013-03-22 16:04:19

Last modified on

2013-03-22 16:04:19

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:24 2018 by

LaTeXML
Chen’s theorem

Chen’s theorem

Theorem

. Every sufficiently large

even integer

n

can be expressed as the sum of two primes

p

+

q

, or the sum of a prime and a

semiprime

p

+

q

⁢

r

, where

p

,

q

and

r

are all distinct primes. “Sufficiently large” could mean

n

>

60

. For example, 62 can be represented as

p

+

q

⁢

r

in seven different ways:

5

+

3

×

19

,

7

+

5

×

11

,

11

+

3

×

17

, etc.

This theorem was proven by Chen Jingrun in 1966 but had to delay publishing his results until 1973 because of political problems in his native China. Chen’s proof has been considered “a highly technical application of sieving methods.” (Eisenstein et al, 2004) Ross simplified Chen’s proof almost a decade later. Still, a summary of the proof can run for dozens of pages. A much shorter, but excessively broad summary that can fit in here goes something like this:

reduction

to sieving, estimation of sieving

functions

, search for

upper bounds

using the Jurkat-Richert theorem, using a

bilinear form

inequality, and joining together of all these results to create a function that counts the number of

representations

of a given number as either

p

+

q

or

p

+

q

⁢

r

, and showing that that function always returns a

positive integer

when the given number is sufficiently large.

The Zumkeller-Lebl conjecture, an attempt to generalize Chen’s theorem to

odd numbers

, and unproven as of 2008, states that sufficiently large

integers

, be they even or odd, can also be represented as

p

+

q

⁢

r

. To represent odd numbers this way, only one of the primes can be 2 (or both

q

=

r

=

2

). Levy’s conjecture, which applies only to odd numbers, has

q

=

2

and

p

and

r

both

odd primes

.

Sequence

A100952 of Sloane’s OEIS lists the known twenty-one small integers that can’t be expressed as specified by the theorem.

References

1

J. R. Chen, “On the representation of a larger even integer as the sum of a prime and the

product

of at most two primes,”

Sci. Sinica

16

(1973), 157 - 176.

2

E. Eisenstein, L. Jain, A. Felix, “A summary of the proof of Chen’s theorem”. Ann Arbor: University of Michigan (2004)

3

P. M. Ross, “On Chen’s theorem that each large

even number

has the form

(

p

1

+

p

2

)

or

(

p

1

+

p

2

⁢

p

3

)

,”

J. London Math. Soc.

10

(1975), 500 - 506

Title

Chen’s theorem

Canonical name

ChensTheorem

Date of creation

2013-03-22 16:19:39

Last modified on

2013-03-22 16:19:39

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

9

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11N05

Generated on Fri Feb 9 15:38:26 2018 by

LaTeXML
Clement’s theorem on twin primes

Clement’s theorem on twin primes

Theorem

. (P. Clement) Given a

prime number

p

,

p

+

2

is also a prime (and

p

and

p

+

2

form a

twin prime

) if and only if

4

⁢

(

p

-

1

)

!

≡

-

4

-

p

(

mod

p

2

+

2

⁢

p

)

.

Richard Crandall and Carl Pomerance see this theorem as “a way to connect the notion of twin-prime pairs with the

Wilson-Lagrange theorem

.”

References

1

Richard Crandall & Carl Pomerance,

Prime Numbers: A Computational Perspective

, 2nd Edition. New York: Springer (2005): 65, Exercise 1.57

Title

Clement’s theorem on twin primes

Canonical name

ClementsTheoremOnTwinPrimes

Date of creation

2013-03-22 17:58:32

Last modified on

2013-03-22 17:58:32

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Theorem

Classification

msc 11N05

Generated on Fri Feb 9 15:38:27 2018 by

LaTeXML
cuban prime

cuban prime

A

cuban prime

is a

prime number

that is a

solution

to one of two different specific equations involving

third powers

of

x

and

y

.

The first of these equations is

p

=

x

3

-

y

3

x

-

y

,

with

x

=

y

+

1

and

y

>

0

. The first few cuban primes from this equation are: 7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919.

The general cuban prime of this kind can be rewritten as

(

y

+

1

)

3

-

y

3

y

+

1

-

y

,

which simplifies to

3

⁢

y

2

+

3

⁢

y

+

1

. This is exactly the general form of a

centered hexagonal number

; that is, all of these cuban primes are centered hexagonal numbers.

This kind of cuban primes has been researched by A. J. C. Cunningham, in a paper entitled ”On quasi-Mersennian numbers”.

As of January 2006 the largest known cuban prime has 65537 digits with

y

=

100000845

4096

, discovered by Jens Kruse Andersen, according to the Prime Pages of the University of Tennessee at Martin.

The second of these equations is

p

=

x

3

-

y

3

x

-

y

,

with

x

=

y

+

2

. It simplifies to

3

⁢

y

2

+

6

⁢

y

+

4

. The first few cuban primes on this form are: 13, 109, 193, 433, 769.

This kind of cuban primes have also been researched by Cunningham, in his book

Binomial

Factorisations

.

The name ”cuban prime” has to do with the rôle cubes (third powers) play in the equations, and has nothing to do with the prime minister of Cuba.

Title

cuban prime

Canonical name

CubanPrime

Date of creation

2013-03-22 16:19:27

Last modified on

2013-03-22 16:19:27

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Related topic

CubeOfANumber

Generated on Fri Feb 9 15:38:29 2018 by

LaTeXML
Cunningham chain

Cunningham chain

Consider the

sequence

of primes 2, 5, 11, 23, 47. Each is twice the previous one plus 1.

When, in a sequence of primes

p

1

,

…

⁢

p

k

each

p

n

=

2

⁢

p

n

-

1

+

1

for

1

<

n

≤

k

(or alternatively, each

p

n

=

p

n

-

1

-

1

2

for

0

<

n

<

k

), the sequence is called a

Cunningham chain

of the

first kind

. In a Cunningham chain of the first kind, all primes except the largest are

Sophie Germain primes

, while all primes except the smallest are

safe primes

. The primes in a Cunningham chain are related to the

Mersenne numbers

thus:

p

n

≡

2

n

-

1

mod

2

n

, where

n

is the prime’s position in the Cunningham chain (except in the case of the chain starting with 2, which is a special case).

In a

Cunningham chain of the

second kind

, the relation among primes is

2

⁢

p

n

-

1

-

1

. An example of a Cunningham chain of the second kind is 1531, 3061, 6121, 12241, 24481.

It is strongly believed that there are infinitely many Cunningham chains of either kind, but this remains to be proven.

External links

http://www.geocities.com/primefan/CunninghamChains.html

PrimeFan’s listing of Cunningham chains

Title

Cunningham chain

Canonical name

CunninghamChain

Date of creation

2013-03-22 16:04:22

Last modified on

2013-03-22 16:04:22

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:31 2018 by

LaTeXML
cyclic number

cyclic number

A

cyclic number

n

is an

integer

which in a given base

b

maintains the same digits after repeated

multiplications

. For example, twice 142857 is 285714; thrice is 428571; four times is 571428; etc. Cyclic numbers are tied to the

full reptend primes

p

thus:

n

=

b

p

-

1

-

1

p

.

From this it follows that the base

b

representation of the number

n

⁢

p

consists of

p

-

1

instances

of the digit

b

-

1

. For

m

⁢

n

with

m

>

p

the

multiples

gradually begin to exhibit greater

variety

in their digits, with the occasional number nearly full of digits

b

-

1

.

In base 10, the first few cyclic numbers are 142857, 588235294117647, 52631578947368421, 434782608695652173913, 344827586206896551724137931, 212765957446808510638297872340425531914893617, etc.

There are no cyclic numbers in

factorial base

or

primorial

base because not every digit can occupy every significant position. For example, take the multiples of 142857 in factorial base: 34222111, 70444300, 115012011, 151234200, 185501311, 232024100, 266251211. The digit 4 can only occur to the left of

d

3

.

Confusingly, Sloane’s OEIS uses this term to refer to the full reptend primes instead of the cyclic numbers they generate.

Title

cyclic number

Canonical

name

CyclicNumber

Date of creation

2013-03-22 16:46:18

Last modified on

2013-03-22 16:46:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:33 2018 by

LaTeXML
full reptend prime

full reptend prime

If for a

prime number

p

in a given base

b

such that

gcd

⁡

(

p

,

b

)

=

1

, the formula

b

p

-

1

-

1

p

gives a

cyclic number

, then

p

is a

full reptend prime

or

long prime

.

The first few base 10 full reptend primes are given by A001913 of Sloane’s OEIS: 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167.

For example, the case

b

=

10

,

p

=

7

gives the cyclic number 142857, thus, 7 is a full reptend prime.

Not all values of

p

will yield a cyclic number using this formula; for example

p

=

13

gives 076923076923. These failed cases will always contain a repetition of digits (possibly several).

The known pattern to this

sequence

comes from

algebraic number theory

, specifically, this sequence is the set of primes

p

such that 10 is a

primitive root

modulo

p

. A

conjecture

of Emil Artin on primitive roots is that this sequence contains about 37

percent

of the primes.

The term

long prime

was used by

John Conway

and Richard Guy in their

Book of Numbers

. Confusingly, Sloane’s OEIS refers to these primes as ”cyclic numbers.”

Title

full reptend prime

Canonical

name

FullReptendPrime

Date of creation

2013-03-22 16:04:50

Last modified on

2013-03-22 16:04:50

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Synonym

long prime

Generated on Fri Feb 9 15:38:34 2018 by

LaTeXML
Kynea number

Kynea number

Given

n

, compute

4

n

+

2

n

+

1

-

1

or

(

2

n

+

1

)

2

-

2

or

4

n

+

∑

i

=

0

n

2

i

.

Any of these formulas gives the

Kynea number

for

n

.

The first few Kynea numbers are 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247 (listed in A093069 of Sloane’s OEIS). Every third Kynea number is

divisible

by 7, thus prime Kynea numbers can’t have

n

=

3

⁢

x

+

2

(except of course for

n

=

2

. The largest Kynea number known to be prime is

(

2

281621

+

1

)

2

-

2

, found by Cletus Emmanuel in November of 2005, using k-Sieve from Phil Comody and OpenPFGW.

Title

Kynea number

Canonical name

KyneaNumber

Date of creation

2013-03-22 16:13:13

Last modified on

2013-03-22 16:13:13

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

4

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11N05

Generated on Fri Feb 9 15:38:36 2018 by

LaTeXML
Numerical verification of the Goldbach conjecture

Numerical verification of the Goldbach conjecture

ABSTRACT:
The Strong Goldbach conjecture, GC, dates back to

1742

. It states that every

even integer

greater than four can be written as the sum of two

prime numbers

. Since then, no one has been able to prove the

conjecture

. The conjecture has been verified to be true for all even integers up to

4.10

18

. In this article, we prove that the conjecture is true for all

integers

, with at least three different ways.
In short, this treaty has as objective show the proof of GC, and presents a new

resolution

to the conjecture. Knowing that, these

infinities

establish other groups of infinities, in a logical way the conviction for the method and idea of proving it, we stand and separate these groups to prove, not only a

sequence

, but the whole embodiment of

arithmetic

properties

called here as groups, as well as its infinity conjectured for centuries.

Keywords:
Goldbach’s Conjecture; Crystallographic group; Cobordism group;

Algebraic number theory

; Multiprime

Theorem

’s; Productoria Table.

AMS Subject Classification:
11N05; 11A41; 11A25; 11Y11; 11P32; 05A10; 11N56; 11D99; 11P99; 11N32; 05A17.

Title

Numerical verification of the

Goldbach conjecture

Canonical

name

NumericalVerificationOfTheGoldbachConjecture

Date of creation

2014-10-25 22:17:31

Last modified on

2014-10-25 22:17:31

Owner

Paulo Fernandesky (1000738)

Last modified by

unlord (1)

Numerical id

5

Author

Paulo Fernandesky (1)

Entry type

Conjecture

Classification

msc 11N05

Classification

msc 11A41

Classification

msc 11A25

Classification

msc 11Y11

Classification

msc 11P32

Classification

msc 05A10

Classification

msc 11N56

Classification

msc 11D99

Classification

msc 11P99

Classification

msc 11N32

Classification

msc 05A17

Generated on Fri Feb 9 15:38:41 2018 by

LaTeXML
prime difference function

prime difference function

The

prime difference function

is an

arithmetic function

for any

positive integer

n

, denoted as

d

n

and gives the

difference

between two consecutive primes

p

n

and

p

n

+

1

:

d

n

≡

p

n

+

1

-

p

n

.

For example:

•

d

1

=

p

2

-

p

1

=

3

-

2

=

1

,

•

d

10

=

p

11

-

p

10

=

31

-

29

=

2

,

•

d

100

=

p

101

-

p

100

=

547

-

541

=

6

,

•

d

1000

=

p

1001

-

p

1000

=

7927

-

7919

=

8

,

•

d

10000

=

p

10001

-

p

10000

=

104743

-

104729

=

14

and so forth.

The first few values of

d

n

for

n

=

1

,

2

,

3

,

…

are

1

,

2

,

2

,

4

,

2

,

4

,

2

,

4

,

6

,

2

,

6

,

4

,

2

,

4

,

6

,

6

,

2

,

6

,

4

,

2

,

…

(

http://www.research.att.com/ njas/

sequences

/eisA.cgi?Anum=001223

OEIS A001223).

Title

prime difference function

Canonical

name

PrimeDifferenceFunction

Date of creation

2013-03-22 12:49:16

Last modified on

2013-03-22 12:49:16

Owner

XJamRastafire (349)

Last modified by

XJamRastafire (349)

Numerical id

6

Author

XJamRastafire (349)

Entry type

Definition

Classification

msc 11N05

Classification

msc 11A25

Classification

msc 11A41

Generated on Fri Feb 9 15:38:43 2018 by

LaTeXML
prime quadruplet

prime quadruplet

A

prime quadruplet

is a set of four

prime numbers

,

p

,

p

+

2

,

p

+

6

,

p

+

8

. In most cases,

p

+

4

is a

multiple

of 15. The only quadruplet for which this is not the case is 5, 7, 11, 13, which overlaps with the quadruplet 11, 13, 17, 19. Sometimes 2, 3, 5, 7 is referred to as a prime quadruplet.

If the

twin prime conjecture

is ever proven it is possible that it might neither prove nor disprove the

prime quadruplet conjecture

.

The sum of the

reciprocals

of the members of the prime quadruplets is Brun’s constant for prime quadruplets,

B

4

≈

0.87058838

.

Title

prime quadruplet

Canonical

name

PrimeQuadruplet

Date of creation

2013-03-22 16:03:02

Last modified on

2013-03-22 16:03:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11N05

Synonym

prime quadruple

Synonym

prime quartet

Generated on Fri Feb 9 15:38:44 2018 by

LaTeXML