Dickson's conjecture
{{Short description|Conjecture about prime numbers}}
In number theory, a branch of mathematics, Dickson's conjecture is the conjecture stated by {{harvs|txt|last=Dickson|authorlink=Leonard Eugene Dickson|year=1904}} that for a finite set of linear forms {{math|a1 + b1n}}, {{math|a2 + b2n}}, ..., {{math|ak + bkn}} with {{math|bi ≥ 1}}, there are infinitely many positive integers {{mvar|n}} for which they are all prime, unless there is a congruence condition preventing this {{harv|Ribenboim|1996|loc=6.I}}. The case k = 1 is Dirichlet's theorem.
Two other special cases are well-known conjectures: there are infinitely many twin primes (n and 2 + n are primes), and there are infinitely many Sophie Germain primes (n and 1 + 2n are primes).
Generalized Dickson's conjecture
{{main|Schinzel's hypothesis H}}
Given n polynomials with positive degrees and integer coefficients (n can be any natural number) that each satisfy all three conditions in the Bunyakovsky conjecture, and for any prime p there is an integer x such that the values of all n polynomials at x are not divisible by p, then there are infinitely many positive integers x such that all values of these n polynomials at x are prime. For example, if the conjecture is true then there are infinitely many positive integers x such that , , and are all prime. When all the polynomials have degree 1, this is the original Dickson's conjecture.
This generalization is equivalent to the generalized Bunyakovsky conjecture and Schinzel's hypothesis H.
See also
References
- {{citation|last=Dickson|first= L. E.|authorlink=Leonard Eugene Dickson|title=A new extension of Dirichlet's theorem on prime numbers|url=https://books.google.com/books?id=i8MKAAAAIAAJ&pg=PA155| journal=Messenger of Mathematics|year=1904|pages = 155–161| volume=33}}
- {{Citation | last1=Ribenboim | first1=Paulo | author1-link=Paulo Ribenboim | title=The new book of prime number records | url=https://books.google.com/books?id=72eg8bFw40kC | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94457-9 | mr=1377060 | year=1996}}
{{Prime number conjectures}}