The (ternary) quadratic form $x^2 + y^2 - z^2$ is universal, meaning that any integer $n$ can be represented as $n = x^2 + y^2 - z^2$ for some integers $x, y, z$.

My question is this: who proved this fact first? I want to know to whom I should credit this fact. The oldest literature I can find is Dickson's 1929 paper "The forms $ax^2+by^2+cz^2$ which represent all integers" in Bulletin of AMS (ProjectEuclid link to paper), where he gives quite a general theorem on universality of all diagonal forms. And I would think that the universality of this specific form can go back further.

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    $\begingroup$ This fact is an immediate consequence from the fact $y^2-z^2$ represents all odd integers, which I imagine goes back centuries. I have no slightest clue when this statement could have first appeared explicitly though $\endgroup$ – Wojowu Oct 14 at 13:24
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    $\begingroup$ In particular, I wouldn't call the first who stated it the "first who proved it". Dickson was maybe the first who addressed the question in this generality. $\endgroup$ – YCor Oct 14 at 13:27
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    $\begingroup$ Dickson's 1929 paper is available at projecteuclid.org/euclid.bams/1183493098 $\endgroup$ – jeq Oct 14 at 13:27
  • $\begingroup$ See Section 5 of a paper of mine published in 2017 available from maths.nju.edu.cn/~zwsun/179b.pdf $\endgroup$ – Zhi-Wei Sun Oct 15 at 1:42

As the others point out, there is no knowing about the earliest this was written down. For example, the notion of regularity of a ternary form is due to Dickson, but universality is an easier concept and could have gone without any name for quite some time.

Worth pointing out that all universal ternaries can be described. Three out of four types are in Dickson's 1939 book, page 161 in Modern Elementary Theory of Numbers; the one type with odd "mixed" coefficients was proved by A. Oppenheim in 1930. Sir Alexander Oppenheim was a student of Dickson and got his Ph.D. in 1930. The dissertation was titled The Minima of Indefinite Quaternary Quadratic Forms


I am having trouble finding this: Quarterly Journal of Mathematics (1930) 179-185. Evidently this is where Oppenheim published a few items. His obituary goes back as far as 1941 ..

The following are representative forms under the action of $SL_3 \mathbb Z$ as follows: given the Hessian matrix $H$ of a quadratic form, a new representative is $P^T H P.$

Taking $N$ odd, while $M$ is any integer, we have universal $$ xy-Mz^2 $$ $$ 2xy - N z^2 $$ $$ 2xy + y^2 - N z^2 $$ $$ 2xy + y^2 - 2N z^2 $$

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  • $\begingroup$ the Oppenheim article is here --- behind a paywall, I do have access, I can email it if so desired. $\endgroup$ – Carlo Beenakker Oct 14 at 16:46
  • $\begingroup$ @CarloBeenakker thanks, got it $\endgroup$ – Will Jagy Oct 14 at 22:39

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