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, M. M. "A Sequence Related to a Conjecture of Schinzel." J. Integer Sequences4, 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.
,
..., 
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.
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).
.
