In 1958, Hasse [10, p. 236], in connection with his work on the 2
-th power character of 2 in the cyclotomic field Q(exp(2πi/2
)), proved that for every prime p ≡ 1 (mod 16) the pair of equations
is always solvable in integers x, u, v, w. Later in 1972 Giudici, Muskat, and Robinson [7, p. 388] showed in their work on Brewer's character sums that Hasse's system is also solvable for primes p ≡ 7 (mod 16). Moreover they also showed [7, p. 345] that for primes p ≡ 1 (mod 5) the pair of equations
is solvable in integers x, u, v, w. In this paper we consider a pair of diophantine equations (involving a prime p and an integer m) which includes, the above two systems as the special cases when m = 2 and m = 5. The system is then used to give criteria for m to be a biquadratic residue modulo p.