Sums of products of powers of given prime numbers
We give the complete solutions of the equations \[ 2^x 3^y+1=2^z+3^w,\quad 2^x 3^y+2^z=3^w+1\quad 2^x 3^y+3^w = 2^z+1 \quad\text{ in integers } x, y, z, w. \] We use this to prove that every large rational number has at most four representations of the form \(2^{\alpha} 3^{\beta}+2^{\gamma}+3^{\delta}\) with \(\alpha, \beta, \gamma, \delta\in\mathbb{Z}\). Finally we prove that, for given integer \(n\) and prime numbers \(p_1,\dots,p_t\) every rational number \(m\) has at most \(C\) representations of the form \(\sum^n_{i=1} p_i^{k_i}\) \((k_1,\dots,k_n\in\mathbb{Z})\) where \(C\) is an integer which depends only on \(n\), \(p_1,\dots,p_t\) and further an extension of this result to algebraic number fields. The proofs of these results are based on effective and ineffective methods from the theory of diophantine approximations. Note: By using a new result of Evertse and Győry the dependence of \(C_ 0\) on \(W\) in Theorem 6 can be relaxed.
scientific article; zbMATH DE number 3979150