A problem from G.E. Andrews's "Number Theory":
Suppose g is a primitive root modulo p (a prime) and m | p-1, where (0<m<p-1). How many integral solutions are there of the congruence
x^m - g "congruent to" 0 (mod p).
Proposed solution:
Assume there is at least one such x.
p ~|g since g is a primitive root.
Therefore p ~| x ^ m and p ~| x.
p ~| x implies x ^ p-1 "congruent to" 1 (Euler's Theorem).
mk= p-1 for some k. (hyp)
x ^ mk "congruent to" 1
x ^ m "congruent to" g. (hyp)
x ^ mk "congruent to" g ^ k. (raise to a power)
g ^ k "congruent to" 1. (Congruence is an equivalence rel.)
k < p-1
g is not a primitive root, since g does not belong to p-1.
Conclusion: there is no x such that x^m "congruent to" g.
Is this conclusion acceptable (or almost acceptable!) ?