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# hypothesis H

In 1958 Schinzel and Sierpinski proposed the following generalization of Dickson's conjecture.

- Conjecture (
Hypothesis H)- Let
kbe 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 primepwhich divides the product f_{1}(m)^{.}f_{2}(m)^{.}. . .^{.}f_{k}(m) for every integerm. Then there exists a positive integernsuch 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*. For each prime

_{i}*p*, let

*w*(

*p*) be the number of solutions to

f_{1}(n)^{.}f_{2}(n)^{.}. . .^{.}f_{k}(n) ≡ 0 (modp)

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. BatemanandR. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers,"Math. Comp.,16(1962) 363-367.MR 26:6139- Ribenboim95 (chapter 6, section IV)
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).

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