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A (positive definite and integral) quadratic form is said to be prime-universal if it represents all primes. Recently, Doyle and Williams [‘Prime-universal quadratic forms 
$ax^2+by^2+cz^2$
and 
$ax^2+by^2+cz^2+dw^2$
’, Bull. Aust. Math. Soc.101 (2020), 1–12] classified all prime-universal diagonal ternary quadratic forms and all prime-universal diagonal quaternary quadratic forms under two conjectures. We classify all prime-universal diagonal quadratic forms regardless of rank, and prove the so-called 67-theorem for a diagonal quadratic form to be prime-universal.
Let 
${{\phi }_{1}},...,{{\phi }_{n}}\,(n\,\ge \,2)$ be nonzero integers such that the equation
$$\sum\limits_{i=1}^{n}{{{\phi }_{i}}x_{i}^{2}\,=\,0}$$
is solvable in integers 
${{x}_{1}},...,{{x}_{n}}$ not all zero. It is shown that there exists a solution satisfying
,
$$0\,<\,\sum\limits_{i}^{n}{\left| {{\phi }_{i}}\left| x_{i}^{2}\,\le \,2 \right|{{\phi }_{1}}...{{\phi }_{n}}\, \right|}$$
and that the constant 2 is best possible.
