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.
