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).
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L. E. "A New Extension of Dirichlet's Theorem on Prime Numbers." Messenger
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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.