Studying a certain problem, I came across the Diophantine equation $x^2 + y^2 + z^2 = 3t^2$. How can I find the solutions to the latter with the further condition that $(x,y,z,t)$ are pairwise coprime? I know that they must be all odds, and that $t$ is the only one that can be a multiple of 3. Do they have other properties? Obviously it admits solutions, such as the trivial $(1,1,1,1)$. Thanks in advance.
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Must $x,y,z,t$ be positive and integer? – ajotatxe Jan 25 '24 at 16:49
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Yes, by symmetry we can restrict the case to all positive integers. – Falcon Jan 25 '24 at 17:22
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Note I found the duplicate through using an Approach0 search (although as a link from its search result of Pythagorean Quadruples:). This other question gives a parametric solution, which doesn't require that $(x,y,z,t)$ be pairwise coprime. However, it does include non-trivial coprime solutions, such as with its $r=1$, $s=t=2$ giving $x=13$, $y=5$, $z=7$ and $t=9$. Checking, we get $169+25+49=243=3\times 81$. – John Omielan Jan 25 '24 at 19:40
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1@JohnOmielan Oh, thanks, I didn't know of approach0 tool. I'm going to use it from now on. – Falcon Jan 25 '24 at 20:39