Schinzel's Hypothesis

If , ...,  are irreducible polynomials with integer coefficients such that no integer  divides , ...,  for all integers , then there should exist infinitely many  such that , ...,  are simultaneously prime.

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

Conroy (2001) verified the conjecture to .

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.