Prime-Universal Quadratic Forms And

A positive-definite diagonal quadratic form a₁x₁²+⋯ +aₙxₙ²\;(a₁,… ,aₙ∈ N) is said to be prime-universal if it is not universal and for every prime p there are integers x₁,… ,xₙ such that a₁x₁²+⋯ +aₙxₙ²=p . We determine all possible prime-universal ternary quadratic forms ax²+by²+cz² and all possible prime-universal 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.