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Mathematics of Computation
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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1981-0628716-9
PII: S 0025-5718(1981)0628716-9
Article copyright: © Copyright 1981 American Mathematical Society