A Hasse-type principle for exponential Diophantine equations and its applications (Q2792344)

The authors consider equations of the shape NEWLINE\[NEWLINEa_1b_{11}^{\alpha_{11}}\cdots b_{1\ell}^{\alpha_{1\ell}}+\cdots +a_kb_{k1}^{\alpha_{k1}}\cdots b_{k\ell}^{\alpha_{k\ell}}=c,\eqno(1)NEWLINE\]NEWLINE to be solved in non-negative integers \(\alpha_{11},\ldots ,\alpha_{k\ell}\), where \(c\) and the \(a_i\) and \(b_{ij}\) are non-zero integers. Such equations are known to have only finitely many non-degenerate solutions, that is, such that the subsums of the left-hand side are non-zero. For \(k=2\) there is a general method to solve such an equation, based on logarithmic form estimates, but for \(k\geq 3\) such a method has not been discovered yet. In the paper under review the authors describe a procedure to solve (1), based on the following Hasse type conjecture:NEWLINENEWLINESuppose Eq. (1) is unsolvable in non-negative integers \(\alpha_{ij}\). Then there exists an integer \(m\geq 2\) such that the congruence NEWLINE\[NEWLINEa_1b_{11}^{\alpha_{11}}\cdots b_{1\ell}^{\alpha_{1\ell}}+\cdots +a_kb_{k1}^{\alpha_{k1}}\cdots b_{k\ell}^{\alpha_{k\ell}}\equiv\, c\,\,\, \pmod m \eqno(2)NEWLINE\]NEWLINE has no solutions in non-negative integers \(\alpha_{ij}\). \newline The author's general strategy to solve (1) is as follows. Suppose one has a list of non-degenerate solutions of (1) and one conjectures that there are no other solutions. Let for instance \(a\) be larger than the \(\alpha_{11}\)-components of all solutions from the list and replace the term \(a_1b_{11}^{\alpha_{11}}\) in (1) by \(a_1'b_{11}^{\alpha_{11}'}\), where \(a_1'=a_1b_{11}^a\) and \(\alpha_{11}'=\alpha_{11}-a\). If one can find a modulus \(m\) such that (2) with \(a_1'b_{11}^{\alpha_{11}'}\) instead of \(a_1b_{11}^{\alpha_{11}}\) is unsolvable in non-negative integers \(\alpha_{11}',\ldots ,\alpha_{k\ell}\), then it follows that the original equation (1) has no solutions with \(\alpha_{11}\geq a\). In this way one can reduce the number of unknowns in (1). This idea is implicit in earlier papers (see for instance [\textit{J. L. Brenner} and \textit{L. L. Foster}, Pac. J. Math. 101, 263--301 (1982; Zbl 0447.10021)]), but there the modulus \(m\) was always chosen on an ad hoc basis for the particular equation of type (1) under consideration. In the present paper, the authors solve various equations of type (1) and give a general strategy to choose \(m\).NEWLINENEWLINEAnother feature of the paper is the following interesting density result, giving evidence for the above conjecture. \newline Fix non-zero integers \(a_i\) and \(b_{ij}\), let \(H_0(x)\) be the number of integers \(c\) with \(|c|\leq x\) such that (1) is unsolvable, and let \(H(x)\) be the number of integers \(c\) with \(|c|\leq x\) such that (1) is unsolvable but (2) is solvable for every integer \(m\geq 2\). Then \(\lim_{x\to\infty} H(x)/H_0(x)=0\). \newline The proof combines various results from earlier papers, but it ultimately depends on the \(p\)-adic subspace theorem of Schmidt and Schlickewei, and on a result of \textit{P. Erdős} et al. [Acta Arith. 58, No. 4, 363--385 (1991; Zbl 0734.11047)], which implies that there is an infinite sequence of integers at which Carmichael's lambda function assumes relatively small values.