Diagonal quadratic forms representing all binary diagonal quadratic forms (Q1696802)

For a positive integer \(n\), a positive definite integral quadratic form is \(n\)-universal if it represents all positive definite integral quadratic forms of rank \(n\). For example, the quinary quadratic form \(x_1^2+x_2^2+x_3^2+x_4^2+ x_5^2\) is \(2\)-universal, by a classical result of \textit{L. J. Mordell} [Q. J. Math., Oxf. Ser. 1, 276--288 (1930; JFM 56.0883.06)]. \textit{B. M. Kim} et al. [Contemp. Math. 249, 51--62 (1999; Zbl 0955.11011)] proved that there exist exactly eleven \(2\)-universal quinary quadratic forms, up to equivalence. In the present paper, the authors consider the weaker condition of being diagonally \(2\)-universal. They prove that a diagonal positive definite integral quadratic form is diagonally \(2\)-universal (that is, represents all diagonal positive definite integral binary quadratic forms) if and only if it represents seven specific binary quadratic forms, and show that there are exactly eighteen diagonal quinary quadratic forms, up to equivalence, having this property.