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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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The probability that a random probable prime is composite
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by Su Hee Kim and Carl Pomerance PDF
Math. Comp. 53 (1989), 721-741 Request permission

Abstract:

Consider a procedure which (1) chooses a random odd number $n \leq x$, (2) chooses a random number b, $1 < b < n - 1$, and (3) accepts n if ${b^{n - 1}} \equiv 1\;\pmod n$. Let $P(x)$ denote the probability that this procedure accepts a composite number. It is known from work of Erdös and the second author that $P(x) \to 0$ as $x \to \infty$. In this paper, explicit inequalities are established for $P(x)$. For example, it is shown that $P({10^{100}}) < 2.77 \times {10^{ - 8}}$ and that $P(x) \leq {(\log x)^{ - 197}}$ for $x \geq {10^{{{10}^5}}}$.
References
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 721-741
  • MSC: Primary 11Y11; Secondary 11A51
  • DOI: https://doi.org/10.1090/S0025-5718-1989-0982368-4
  • MathSciNet review: 982368