Functiones et Approximatio Commentarii Mathematici

Lucas non-Wieferich primes in arithmetic progressions

Sudhansu Sekhar Rout

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In this note, we define the Lucas Wieferich primes which are an analogue of the famous Wieferich primes. Conditionally there are infinitely many non-Wieferich primes. We prove under the assumption of the $abc$ conjecture for the number field $\mathbb{Q}(\sqrt{\Delta})$ that for fixed positive integer~$M$ there are at least $O((\log x/\log \log x)(\log \log \log x)^{M})$ many Lucas non-Wieferich primes $p \equiv 1(mod k)$ for any fixed integer $k\geq 2$.

Article information

Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 167-175.

First available in Project Euclid: 29 November 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 11B25: Arithmetic progressions [See also 11N13] 11A41: Primes

Lucas-Wieferich primes arithmetic progressions $abc$ conjecture


Rout, Sudhansu Sekhar. Lucas non-Wieferich primes in arithmetic progressions. Funct. Approx. Comment. Math. 60 (2019), no. 2, 167--175. doi:10.7169/facm/1709.

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