PRIME-UNIVERSAL QUADRATIC FORMS ax2 + by2 + cz2 AND ax2 + by2 + cz2 + dw2

A positive-definite diagonal quadratic form a1x2 1 + + anx2 n (a1; : : :; an 2 N) is said to be prime-universal if it is not universal and for every prime p there are integers x1; : : :; xn such that a1x2 1 + + anx2 n = p. We determine all possible prime-universal ternary quadratic forms ax2 + by2 + cz2 and all possible primeuniversal quaternary quadratic forms ax2 + by2 + cz2 + dw2. The prime-universal ternary forms are completely determined. The prime-universal quaternary forms are determined subject to the validity of two conjectures. We make no use of a result of Bhargava concerning quadratic forms representing primes which is stated but not proved in the literature.