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In 1958 Schinzel and Sierpinski proposed the following
generalization of Dickson's conjecture.
 Conjecture (Hypothesis H)
 Let k be a positive integer and let
f_{1}(x), f_{2}(x),
. . ., f_{k}(x) be irreducible
polynomials with integral coefficients and positive leading
coefficients. Assume also that there is not a prime
p which divides the product
f_{1}(m)^{.}f_{2}(m)^{.}
. . . ^{.} f_{k}(m) for every
integer m. Then there exists
a positive integer n such that
f_{1}(n), f_{2}(n),
. . ., f_{k}(n) are all primes.
If there one such n making these polynomials
all prime, then there are infinitely many
such n. So, for example, this conjecture implies there are infinitely many primes of the form
n^{2}+1.
Hypothesis H was quantified as follows by Bateman and
Horn in 1962. Let d_{i} be the degree of f_{i}. For each prime p, let w(p) be the number of solutions to
f_{1}(n)^{.}f_{2}(n)^{.} . . . ^{.} f_{k}(n) = 0
(mod p)
then the expected number of values of n less than
N for which f_{1}(n), f_{2}(n), . . ., f_{k}(n) are simultaneously prime, is
.
References:
 BH62
 P. T. Bateman and R. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math. Comp., 16 (1962) 363367. MR 26:6139
 Ribenboim95 (chapter 6, section IV)
 P. Ribenboim, The new book of prime number records, 3rd edition, SpringerVerlag, New York, NY, 1995. pp. xxiv+541, ISBN 0387944575. 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 seventyfive pages.]
 SS58
 A. Schinzel and W. Sierpinski, "Sur certaines hypotheses concernment les nombres premiers," Acta. Arith., 4 (1958) 185208. Erratum 5 (1958).
