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Skolem’s conjecture confirmed for a family of exponential equations. II. (English) Zbl 1465.11085

In 1937, T. Skolem conjectured that if an exponential Diophantine equation is not solvable, then it is not solvable modulo an appropriate modulus. In this paper, showing Skolem’s conjecture is valid for equations of the form \(x^n-by_1^{k_1}\dots y_l^{k_l}=\pm 1\), where \(b,x,y_1,\dots y_l\) are fixed integers and \(n,k_1,\dots k_l\) are non-negative integral unknowns, the authors give the following result:
Theorem 1. Suppose that \(b,x,y_1,\dots , y_l\) are integers. Then there exists a modulus \(m\) such that the congruence \[x^n-by_1^{k_1}\dots y_l^{k_l}\equiv \pm 1 \pmod{m} \] has precisely the same solutions in non-negative integers \(n,k_1,\dots,k_l\) as the equation \[x^n-by_1^{k_1}\dots y_l^{k_l}= \pm 1. \] Theorem 1 extends the result of L. Hajdu and R. Tijdeman [Acta Arith. 192, No. 1, 105–110 (2020; Zbl 1450.11027)] proves that Skolem’s conjecture holds for the Catalan equation \(x^n-y^k=1\) for fixed bases \(x,y\). Theorem 1 can be applied for another well-known equation, namely for \(\dfrac{x^n-1}{x-1}=y^k\) in integers \(x,y,n,k\) with \(x > 1\), \(y > 1\), \(n > 2\), \(k > 1\).
In the proofs, they use skillfully elementary methods of number theory.

MSC:

11D61 Exponential Diophantine equations
11D79 Congruences in many variables

Citations:

Zbl 1450.11027
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References:

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