We study the multidimensional distribution of the power generator of pseudorandom numbers modulo a high power of a fixed prime number. These results complement some recently obtained results about the power generator modulo a product of two distinct primes in which case the generator is of great value for many cryptographic applications. The case of a prime power modulus, although it does not have any immediate cryptography related applications, may still be of interest for other applications which require quality pseudorandom numbers. Moreover, in this case new effects arise which allow us to apply some recent bounds for exponential sums with sparse polynomials to study the multidimensional distribution. In the case of moduli which are the product of two primes such results are known only for power generators with small exponents.