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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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Sharpening “Primes is in P” for a large family of numbers
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by Pedro Berrizbeitia PDF
Math. Comp. 74 (2005), 2043-2059 Request permission

Abstract:

We present algorithms that are deterministic primality tests for a large family of integers, namely, integers $n \equiv 1\pmod 4$ for which an integer $a$ is given such that the Jacobi symbol $(\frac {a}{n})= -1$, and integers $n \equiv {-1} \pmod 4$ for which an integer $a$ is given such that $(\frac {a}{n})= (\frac {1-a}{n})=-1$. The algorithms we present run in $2^{-\min (k,[2 \log \log n])} \tilde {O}((\log n)^6)$ time, where $k = \nu _2(n-1)$ is the exact power of $2$ dividing $n-1$ when $n \equiv 1\pmod 4$ and $k = \nu _2(n+1)$ if $n \equiv \ -1\pmod 4$. The complexity of our algorithms improves up to $\tilde {O}((\log n)^4)$ when $k \geq [2 \log \log n]$. We also give tests for a more general family of numbers and study their complexity.
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Additional Information
  • Pedro Berrizbeitia
  • Affiliation: Departamento de Matemáticas Puras y Aplicadas, Universidad Simón Bolívar, Caracas, Venezuela
  • Email: pedrob@usb.ve
  • Received by editor(s): February 4, 2003
  • Received by editor(s) in revised form: June 12, 2004
  • Published electronically: April 11, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 2043-2059
  • MSC (2000): Primary 11Y11, 11Y16
  • DOI: https://doi.org/10.1090/S0025-5718-05-01727-8
  • MathSciNet review: 2164112