On ...-Adic Representations Attached to Modular Forms.
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.
We prove that almost all positive even integers can be represented as with for . As a consequence, we show that each sufficiently large odd integer can be written as with for .