Consider an exponential equation\[F(\mathbf{x})=\sum_{i=1}^r a_i^\cdot b_{i1}^{x_{i1}}\cdots b_{i,\ell_i}^{x_{i,\ell_i}}=0\]to be solved in tuples of positive integers \(\mathbf{x}=(x_{i,j}:\, i=1,\ldots , r,\, j=1,\ldots , \ell_i)\in\mathbb{Z}^{\ell_1+\cdots +\ell_r}\) where the \(a_i\) are non-zero integers and the \(b_{i,j}\) integers with absolute value \(>1\). A Skolem-type conjecture asserts that there is a positive integer \(m\) such that this equation has the same set of solutions as the congruence equation \(F(\mathbf{x})\equiv 0\pmod m\). While this is false in general, it is believed that under suitable coprimality conditions imposed on the \(a_i\) and \(b_{i,j}\) and non-degeneracy conditions on the solutions it might be true. Skolem-type conjectures have been proved in a couple of cases. The authors prove another case. Their main result is as follows.
Theorem. Let \(a,c,t,b_1,\ldots ,b_{\ell}\) be non-zero integers with \(|b_i|>1\) for \(i=1,\ldots ,t\) and
\(\gcd (a,c,tb_1\cdots b_{\ell})=1\) and let \(\varepsilon\in\{ -1,1\}\). Further, let \(f\) be any monotone non-decreasing real function. Then there exists a modulus \(m\), which can be effectively calculated in terms of \(a,c,t,b_1,\ldots ,b_{\ell}\) and \(f\), such that the congruence\[a^n+tb_1^{k_1}\cdots b_{\ell}^{k_{\ell}}\equiv \varepsilon c^n \pmod m,\ \ n\leq f(k_1,\ldots ,k_{\ell})\]has the same solutions in positive integers \(n,k_1,\ldots k_{\ell}\) as the equation\[a^n+tb_1^{k_1}\cdots b_{\ell}^{k_{\ell}}=\varepsilon c^n,\ \ n\leq f(k_1,\ldots ,k_{\ell}).\]It should be observed that using Baker-type results on linear forms in ordinary and \(p\)-adic logarithms, one can effectively compute a number \(C\) such that the solutions of the latter equation, even without the condition \(n\leq f(k_1,\ldots , k_{\ell})\), satisfy \(n,k_1,\ldots ,k_{\ell}<C\). This implies the above theorem at once. However, the authors give an elementary proof of their theorem. Their main tool is Zsigmondy’s theorem [
K. Zsigmondy, Monatsh. Math. Phys. 3, 265–284 (1892;
JFM 24.0176.02)], which asserts that if \(a,c\) are coprime integers with \(|ac|>1\), then apart from at most four integers \(n\geq 2\), the number \(a^n-c^n\) has a prime factor \(p\) that does not divide \(a^r-c^r\) for \(r=1,\ldots ,n-1\).
For Part II, see [
A. Bérczes et al., Acta Arith. 197, No. 2, 129–136 (2021;
Zbl 1465.11085)].
By taking \(\ell =1\), \(n=k_1\), \(f(x)=x\), the authors deduce that if \(a,b,c,t\) are non-zero integers with \(\gcd (a,tb,c)=1\), \(|b|>1\) and if \(\varepsilon\in\{ -1,1\}\), then there exists a modulus \(m\), effectively computable in terms of \(a,b,c,t\), such that the congruence \(a^n+tb^n\equiv \varepsilon c^n\pmod m\) has the same solutions in positive integers \(n\) as the equation \(a^n+tb^n=\varepsilon c^n\).
