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A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.

In 1911, Dubouis determined all positive integers that are represented by a sum of k positive squares for any k ≥ 4 . In this article, we generalize Dubouis' result to the binary case. We determine all binary forms that are represented by a sum of k nonzero squares for any k ≥ 5 .

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=Sa={an2∣n≥2} or S=Sa,b={an2+b∣n∈ℤ}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many Sa-universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S1-universal quadratic forms not representing 1.