Skolem's conjecture confirmed for a family of exponential equations, III

Hajdu, Lajos; Luca, Florian; Tijdeman, R.

doi:10.1016/j.jnt.2021.01.013

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“…, k ℓ are non-negative integral unknowns. This result extends the above mentioned theorem of Hajdu and Tijdeman [7] that covers the case b = ℓ = 1 and one of x, y 1 is a prime. Note that in particular, our present results imply that Skolem's conjecture holds for the Catalan equation x n − y k = 1 for fixed bases x, y.…”

Section: Introductionsupporting

confidence: 85%

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Skolem’s conjecture confirmed for a family of exponential equations, II

2021

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According to Skolem's conjecture, if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriately chosen modulus. Besides several concrete equations, the conjecture has only been proved for rather special cases. In this paper we prove the conjecture for equations of the form x n − by k1 1 . . . y k ℓ ℓ = ±1, where b, x, y 1 , . . . , y ℓ are fixed integers and n, k 1 , . . . , k ℓ are non-negative integral unknowns. This result extends a recent theorem of Hajdu and Tijdeman.

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Section: Introductionsupporting

confidence: 85%

“…Recently, Hajdu and Tijdeman [7] proved that Skolem's conjecture is valid for the Catalan equation x n − y k = 1, if x and y are fixed positive integers one of which is prime and n, k are non-negative integer variables. Further related theoretical and numerical results can be found e.g.…”

Section: Introductionmentioning

confidence: 99%