Experimental Mathematics

Euclid Prime Sequences over Unique Factorization Domains

Nobushige Kurokawa and Takakazu Satoh

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Abstract

The proof by Euclid that there exist infinitely many prime numbers is well known. The proof involves generating prime numbers that do not belong to a given finite set of primes, and one may ask whether all prime numbers can be obtained by this method. Daniel Shanks gave a heuristic argument that suggests that the answer is affirmative. Despite recent advances in computational number theory, numerical examples do not seem to make this conjecture convincing. We reformulate the problem in polynomial rings over finite fields and prove that in some explicitly characterized cases, Shanks’s argument does not hold. On the other hand, we have performed numerical computations that suggest that except for the above cases, Shanks’s conjecture is true.

Article information

Source
Experiment. Math., Volume 17, Issue 2 (2008), 145-152.

Dates
First available in Project Euclid: 19 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.em/1227118967

Mathematical Reviews number (MathSciNet)
MR2433881

Zentralblatt MATH identifier
1216.11107

Subjects
Primary: 11T55: Arithmetic theory of polynomial rings over finite fields 11A51: Factorization; primality 13F15: Rings defined by factorization properties (e.g., atomic, factorial, half- factorial) [See also 13A05, 14M05] 13P05: Polynomials, factorization [See also 12Y05]

Keywords
Prime sequences polynomial factorization the Shanks conjecture

Citation

Kurokawa, Nobushige; Satoh, Takakazu. Euclid Prime Sequences over Unique Factorization Domains. Experiment. Math. 17 (2008), no. 2, 145--152. https://projecteuclid.org/euclid.em/1227118967


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