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.

中文翻译：

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PRIME-UNIVERSAL 对角二次型

一个（正定和积分）二次形式被称为普世的如果它代表所有素数。最近，Doyle 和 Williams ['Prime-universal quadratic forms$ax^2+乘^2+cz^2$和$ax^2+乘^2+cz^2+dw^2$',公牛。澳大利亚。数学。社会党。101(2020), 1-12] 在两个猜想下分类所有素泛对角三元二次形式和所有素泛对角四元二次形式。我们对所有素泛对角二次型进行分类，而不考虑秩，并证明对角二次型的所谓 67 定理是素泛的。