What should be assumed about the integral polynomials in order that the solvability of the congruence for sufficiently large primes p implies the solvability of the equation in integers x? We provide some explicit characterizations for the cases when are binomials or have cyclic splitting fields.
Grosswald’s conjecture is that g(p), the least primitive root modulo p, satisfies g(p) ≤ √p - 2 for all p > 409. We make progress towards this conjecture by proving that g(p) ≤ √p -2 for all and for all .
In the first part, we assign to each positive integer a digraph whose set of vertices consists of elements of the ring with the addition and the multiplication operations modulo and for which there is a directed edge from to if and only if . Associated with are two disjoint subdigraphs: and whose union is The vertices of are coprime to and the vertices of are not coprime to In this part, we study the structure of in detail. In the second part, we investigate the zero-divisor...