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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Primes differing by a fixed integer

Authors: W. G. Leavitt and Albert A. Mullin
Journal: Math. Comp. 37 (1981), 581-585
MSC: Primary 10L10; Secondary 10H15
MathSciNet review: 628716
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Abstract: It is shown that the equation $( \ast )\;{(n - 1)^2} - \sigma (n)\phi (n) = {m^2}$ is always solvable by $n = {p_1}{p_2}$ where ${p_1},{p_2}$ are primes differing by the integer m. This is called the "Standard" solution of $( \ast )$ and an m for which this is the only solution is called a "$^\ast$-number". While there are an infinite number of non $^\ast$-numbers there are many (almost certainly infinitely many) $^\ast$-numbers, including $m = 2$ (the twin prime case). A procedure for calculating all non $^\ast$-numbers less than a given bound L is devised and a table is given for $L = 1000$.

References [Enhancements On Off] (What's this?)

  • S. A. Sergušov, On the problem of prime-twins, Jaroslav. Gos. Ped. Inst. Učen. Zap. Vyp. 82 Anal. i Algebra (1971), 85–86 (Russian). MR 0480384

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Article copyright: © Copyright 1981 American Mathematical Society