Some results concerning the exponential Diophantine equation $$(a^n-1)(b^m-1) = x^2$$ ( a n - 1 ) ( b m - 1 ) = x 2

Zahra Ameur (), Rachid Boumahdi () and Tarek Garici ()
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Zahra Ameur: USTHB
Rachid Boumahdi: USTHB
Tarek Garici: USTHB

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 613-622

Abstract: Abstract Let $$a,b>1$$ a , b > 1 be two given unequal integers. In this paper, we examine, under some assumptions on the residue of a modulo an odd prime factor p of b, the Diophantine equation of the title with positive integer unknowns n, m and x. For example, we show that it has no solution when $$p\equiv \pm 3 \pmod 8$$ p ≡ ± 3 ( mod 8 ) , $$a\equiv p-1 \pmod {2p}$$ a ≡ p - 1 ( mod 2 p ) and $$ b\equiv 3 \pmod 4$$ b ≡ 3 ( mod 4 ) . We also give a necessary and sufficient condition for the existence of solutions with $$n=m$$ n = m , when $$p\equiv \pm 3 \pmod 8$$ p ≡ ± 3 ( mod 8 ) , $$a \equiv -1 \pmod p$$ a ≡ - 1 ( mod p ) and when $$p=17 $$ p = 17 , $$a^2 \equiv -1 \pmod {17}$$ a 2 ≡ - 1 ( mod 17 ) .

Keywords: Exponential Diophantine equation; Pell equation; Legendre symbol; Chebyshev polynomials; 11D41; 11D61 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00391-5

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