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Mathematics of Computation

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Gaps between integers
with the same prime factors

Authors: Todd Cochrane and Robert E. Dressler
Journal: Math. Comp. 68 (1999), 395-401
MSC (1991): Primary 11N25, 11N05
MathSciNet review: 1613691
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Abstract: We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers $a <c$ is $\gg a^{1/2 - \epsilon }$, and it is shown that this lower bound is best possible. Dressler's conjecture is verified for values of $a$ and $c$ up to $7\cdot 10^{13}$.

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Additional Information

Todd Cochrane
Affiliation: Kansas State University, Manhattan KS 66506, U. S. A.

Robert E. Dressler
Affiliation: Kansas State University, Manhattan KS 66506, U. S. A.

Keywords: Primes, abc
Received by editor(s): February 24, 1996
Received by editor(s) in revised form: October 7, 1996
Additional Notes: The authors wish to thank the referee for his/her helpful comments, which inspired the addition of Theorem 2 and the Example to the paper.
Article copyright: © Copyright 1999 American Mathematical Society