On a few Diophantine equations, in particular, Fermat's Last Theorem.
The number of solutions of the congruence in the box is estimated from below in the best possible way, provided for all i,j either or or .
Given an integer , let be pairwise coprime integers , a family of nonempty proper subsets of with “enough” elements, and a function . Does there exist at least one prime such that divides for some , but it does not divide ? We answer this question in the positive when the are prime powers and and are subjected to certain restrictions.We use the result to prove that, if and is a set of three or more primes that contains all prime divisors of any number of the form for...