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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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Primes differing by a fixed integer
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by W. G. Leavitt and Albert A. Mullin PDF
Math. Comp. 37 (1981), 581-585 Request permission

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
  • 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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Additional Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Math. Comp. 37 (1981), 581-585
  • MSC: Primary 10L10; Secondary 10H15
  • DOI: https://doi.org/10.1090/S0025-5718-1981-0628716-9
  • MathSciNet review: 628716