Mersenne numbers which are products of two Pell numbers

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Alan M., Şimşek Alan K.

BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA, vol.28, no.2, 2022 (Peer-Reviewed Journal)

• Publication Type: Article / Article
• Volume: 28 Issue: 2
• Publication Date: 2022
• Doi Number: 10.1007/s40590-022-00430-x
• Journal Name: BOLETIN DE LA SOCIEDAD MATEMATICA MEXICANA
• Keywords: Pell numbers, Mersenne numbers, Diophantine equations, Linear forms in logarithms, FIBONACCI NUMBERS, EQUATIONS, SUMS, REPDIGITS, POWERS
• Open Archive Collection: AVESIS Open Access Collection
• Yıldız Technical University Affiliated: Yes

Abstract

Let (P-n)(n >= 0) be the Pell sequence defined by P-0 = 0, P-1 = 1 and Pn+2 2P(n+1) + P-n for n >= 0 and M-k + 2(k) - 1 be the k-th Mersenne number for k >= 1: We show that the Diophantine equation PnPm + M-k with m <= n has only the unique positive integer solution (n, m, k) + (1; 1; 1).