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Schinzel's Hypothesis


If f_1(x), ..., f_s(x) are irreducible polynomials with integer coefficients such that no integer n>1 divides f_1(x), ..., f_s(x) for all integers x, then there should exist infinitely many x such that f_1(x), ..., f_s(x) are simultaneously prime.

If Schinzel's hypothesis is true, then it follows that all positive integers n can be represented in the form n=(p+1)/(q+1) with p and q prime. In addition, it would follow that there are an infinite number of numbers n such that sigma(d(n))=d(sigma(n)), where d(n) is the number of divisors of n and sigma(n) is the sum of divisors, since the conjecture implies that there are infinitely many primes p for which (p^2+p+1)/3 is prime, for such p, d(sigma(p^2))=d(p^2+p+1)=4 and sigma(d(p^2))=sigma(3)=4, so p^2 is in the sequence (D. Hickerson, pers. comm., Jan. 24, 2006).

Conroy (2001) verified the conjecture to n=10^9.


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References

Conroy, M. M. "A Sequence Related to a Conjecture of Schinzel." J. Integer Sequences 4, No. 01.1.7, 2001. http://www.cs.uwaterloo.ca/journals/JIS/VOL4/CONROY/conroy.html.Dickson, L. E. "A New Extension of Dirichlet's Theorem on Prime Numbers." Messenger Math. 33, 155-161, 1904.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.Schinzel, A. and Sierpiński, W. "Sur certaines hypothèses concernant les nombres premiers. Remarque." Acta Arithm. 4, 185-208, 1958.Schinzel, A. and Sierpiński, W. Erratum to "Sur certains hypothèses concernant les nombres premiers." Acta Arith. 5, 259, 1959.

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Schinzel's Hypothesis

Cite this as:

Weisstein, Eric W. "Schinzel's Hypothesis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchinzelsHypothesis.html

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