Dickson's conjecture

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Dickson's conjecture is an extension of Dirichlet's theorem, also called the Dirichlet prime number theorem (for linear forms, i.e. arithmetic progressions), to a finite set of arithmetic progressions.

Conjecture (Dickson's conjecture, 1904). (Dickson)

For a finite set of linear forms

{a1n + b1, a2 n + b2, ..., ak n + bk},

with 

ai   ≥   1

and 

(ai, bi ) = 1

for 

i = 1

to 

k

, there are infinitely many positive integers 

n

for which they are all prime, unless there is a congruence condition preventing this (Ribenboim 1996, 6.I), i.e. there is a prime 

p

which divides 

∏ k i  = 1  (ai n + bi )

for all 

n

. The case 

k = 1

is Dirichlet's theorem.

Special cases of Dickson's conjecture

The following conjectures are all special cases of Dickson's conjecture:

• the prime k-tuple conjecture, which subsumes:

• de Polignac's conjecture: if the set of linear forms is 

  {n, n + 2 k}, k   ≥   1

  , which itself subsumes:

• the twin prime conjecture: if the set of linear forms is 

  {n, n + 2}

  ;
• the cousin prime conjecture: if the set of linear forms is 

  {n, n + 4}

  ;
• ...

• the conjecture that there are infinitely many Sophie Germain primes: if the set of linear forms is 

  {n, 2n + 1}

  ;
• the conjecture that for each positive integer 

  k

  , there is an arithmetic sequence of 

  k

  primes.

Generalizations of Dickson's conjecture

• Schinzel's hypothesis H (for a finite set of integer polynomials with arbitrary degree) (This is an extension on the earlier Bunyakovsky conjecture, for a single integer polynomial with arbitrary degree.)

Sequences

A088250 Smallest number 

k

such that 

k r + 1

is prime for all 

r = 1

to 

n, n   ≥   1

.

 

{ 1, 1, 2, 330, 10830, 25410, 512820, 512820, 12960606120, 434491727670, 1893245380950, 71023095613470, 878232256181280, ... }

References

• Dickson, L. E. (1904). “A new extension of Dirichlet's theorem on prime numbers”. Messenger of mathematics 33: pp. 155–161. https://books.google.com/books?id=i8MKAAAAIAAJ&pg=PA155. 

• Ribenboim, Paulo (1996). The new book of prime number records. Berlin, New York: Springer-Verlag. ISBN 978-0-387-94457-9. https://books.google.com/books?id=72eg8bFw40kC.