Abstract:

In this article, we determine all solutions to the equation of the form 9(5^x )-p^y=z^4, where p is a prime number and x, y, z are nonnegative integers. By employing elementary techniques involving congruences, we establish that the equation admits solutions only when both x and y are odd and p\equiv p\equiv 29\text{ or }149\pmod{240}, or when (x, y, z, p) = (0, 2, 0, 3) or (0, 3, 1, 2).