We derive closed-form counting formulas for distinct non-vanishing representations of prime powers by sums of three and four squares. The formulas for even powers of primes depend only on the residue classes of primes modulo 8 (modulo 24) for three squares (four squares). The formulas for odd powers of primes depend via the theorem of Gauss on class numbers of binary quadratic forms with negative discriminants.

sum of squares, prime powers, Pythagorean quadruple, Pythagorean quintuple, theorem of Gauss, class number, counting formula.