While I was reading a paper on number theory, there was a claim which wasn't prove there and I couldn't find a way to justify it. The claim is as follows
For a prime $p$, when $p\nmid a$, the number of incongruent solutions to $x^3\equiv a \mod p^3$ is $O(1)$.
Is there an elementary proof of this?
reference to the paper (page 28): Brüdern, Jörg. A note on cubic exponential sums. Seminaire de Theorie des Nombres, Paris, 199091, 2334, Progr. Math., 108, Birkhäuser Boston, Boston, MA, 1993.
At first use this fact that the congruent $$ x^3\equiv a\pmod p $$ Has at most three solutions, hence the congruent $$ x^3\equiv a\pmod {p^3} $$ has a bounded numbers (independent of $p$) of solutions according to Hensel's lemma.