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The smallest Prime Value of xn + a

Published online by Cambridge University Press:  20 November 2018

Kevin S. McCurley
Kevin S. McCurley*
Affiliation:
Michigan State University, East Lansing, Michigan University of Southern California, Los Angeles, California
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Let a and n be positive integers such that f(x) = xn + a is irreducible over the integers. A conjecture made by Bouniakowsky [4] in 1857 would imply that there exist infinitely many integers x such that f(x) is prime. An even stronger conjecture of Bateman and Horn [1, 2] would imply that

where π(x;f) is the number of integers m with 0 ≧ m ≧ x for which f(m) is prime, and

where w(p) is the number of solutions of the congruence

Except for the trivial case n = 1, neither of these conjectures has ever been resolved.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 4 , 01 August 1986 , pp. 925 - 936
Copyright
Copyright © Canadian Mathematical Society 1986

References

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