# Dickson's conjecture

Dickson conjectured in 1904 that given a family of linear functions with integer
coefficients *a*_{i} > 1 and *b*_{i}:

a_{1}n+b_{1},a_{2}n+b_{2},a_{k}n+b_{k},

then there are infinitely many integers *n* > 0 for
which these are *simultaneously* prime unless they
"obviously cannot be" (that is, unless there is
a prime *p* which divides the product of these
*for all* *n*). This is now called **Dickson's Conjecture**.

Many conjectures follow from Dickson's conjecture.
For example, if the functions are *n* and
*n*+2, then Dickson's conjecture implies
the twin prime conjecture. If the functions are
*n* and 2*n*+1, then we have the conjecture
that there are infinitely many Sophie Germain primes.
The prime k-tuple conjecture is also a special case
of Dickson's conjecture, as is the conjecture that
for each positive integer *n*, there is an
arithmetic sequence of *n* primes. Finally,
if Dickson's conjecture is true, then there are
infinitely many composites Mersenne numbers as
well as infinitely many Carmichael numbers with
just three prime factors.

Schinzel and Sierpinski extended Dickson's conjecture into the analogous Hypothesis H for integer polynomials with arbitrary degree.

Dickson's conjecture can be heuristically
quantified as follows. Let w(*p*) be the number of solutions
to

(a_{1}n+b_{1})(a_{2}n+b_{2})^{.}...^{.}(a_{k}n+b_{k}) = 0 (modp).

Then the expected number of positive integers *n* less
than *N* which yield *k* simultaneous
primes

a_{1}n+b_{1},a_{2}n+b_{2},a_{k}n+b_{k},

is conjectured to be asymptotic to

where the products are taken over the set of all primes
*p*.

If we replace "there are **infinitely many
integers** *n* > 0 for which these are
simultaneously prime" in Dickson's conjecture with
"there is **an integer** *n* > 0
for which these are simultaneously prime," then
we appear to weaken the conjecture. But it is easy to
show these two conjectures are equivalent!

Another important conjecture that follows from Dickson's
conjecture is that if *a*_{1}
< *a*_{2} < ... < *a*_{k} are nonzero integers for which
there is no prime dividing the product

(x+a_{1})(x+a_{2})^{.}...^{.}(x+a_{k})

for all integers *n*, then there are
infinitely many positive integers *n*
for which

x+a_{1},x+a_{2}, . . .x+a_{k},

are *consecutive* primes.

This form of Dickson's conjecture implies that for
each positive integer *n*, there are infinitely many
arithmetic sequence of *n* consecutive primes.
It also implies Polignac's conjecture, and ...

**Related pages** (outside of this work)

**References:**

- Ribenboim95 (chapter 6)
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, New York, NY, 1995. pp. xxiv+541, ISBN 0-387-94457-5.MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]- SS58
A. SchinzelandW. Sierpinski, "Sur certaines hypotheses concernment les nombres premiers,"Acta. Arith.,4(1958) 185-208. Erratum5(1958).