Exponential congruences (Q2707577)

This short survey presents old and new results on a conjecture of Skolem (1937): NEWLINENEWLINENEWLINEConjecture. Let \(K\) be a finite extension of the field of rationals and \(\beta _{hi}\in K\), \(\alpha _{hij}\in K^*\). If the system of congruences NEWLINE\[NEWLINE \sum_{h=1}^g \prod_{j=1}^k \alpha _{hij}^{x_j}\equiv 0 \pmod {\mathfrak m} \quad \text{(\(1\leq i\leq l\))} NEWLINE\]NEWLINE is solvable for all ideals \({\mathfrak m}\) of \(K\) then the corresponding system of equations is solvable in integers. For the sake of simplicity we quote only one result: NEWLINENEWLINENEWLINEProposition. Let \((F_n)\) be the Fibonacci sequence. If a congruence \(F_n \equiv a \pmod p\) is solvable for almost all primes \(p\), then \(a=F_k\) for some \(k\in {\mathbb Z}\). (Here almost all primes means all primes except a set of zero Dirichlet density.) NEWLINENEWLINENEWLINEand one open question:NEWLINENEWLINENEWLINEProblem. Assume that \(F_{3n} \equiv a\pmod p\) is solvable for all primes \(p\). Is it true that \(a=F_{3k}\)?NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].