## Experimental Mathematics

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

### Euclid Prime Sequences over Unique Factorization Domains

Nobushige Kurokawa and Takakazu Satoh

#### 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