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Probably duplicate but I don't find: I'd like to solve the diophantine equation

$$x^2+y^2+z^2=a^2$$

which has solutions, by exemple $1^2+2^2+2^2=3^2$ or $2^2+3^2+6^2=7^2$. Every such solution gives a rational point on the unit sphere.

Is there a complete description of the solutions such as for pythagorician triplet?

Gabriel Soranzo
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1 Answers1

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We'll start with Pythagorean triples to see the pattern. The complete rational solution to $x_1^2+x_2^2 = y_1^2$ has the form,

$$((a^2-b^2)t)^2+(2abt)^2 = ((a^2+b^2)t)^2\tag{1}$$

where $t$ is a scaling factor.

Proof: For any solution where $x_1+y_1 \neq 0$ , one can always find rational {$a,b,t$} using the formulas $a,b,t = x_1+y_1,\; x_2,\; \frac{1}{2(x_1+y_1)}$.

Similarly, for $x_1^2+x_2^2+x_3^2 = y_1^2$, it is,

$$((a^2-b^2-c^2)t)^2+(2abt)^2+(2act)^2 = ((a^2+b^2+c^2)t)^2\tag{2}$$

Proof: One can always find rational {$a,b,c,t$} using $a,b,c,t = x_1+y_1,\; x_2,\; x_3,\; \frac{1}{2(x_1+y_1)}$.

and so on for $n$ squares. See also Sums of Three Squares for more.

Tito Piezas III
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  • Bautiful identity! It seems to me that the proof doesn't cover the integer case. Strange that for the classic case $x_1^2+x_2^2=y_1^2$ there's an "geometric" approach with $\mathbb{Z}[i]$ but there don't seem to be something similar for this case but we have still a good description of the solutions. – Gabriel Soranzo Jan 22 '14 at 14:54
  • I think there is geometric approach. Consider that $x^2+y^2=1$ describes a circle. Demjanenko studied $a^4+b^4+c^2=1$ as a pencil of conics, and Elkies specialized this to $a^4+b^4+c^4=1$ as a pencil of curves of genus one. – Tito Piezas III Jan 22 '14 at 20:41
  • I'm curious… for which $n$ does the complete rational solution [in rational parameters] have the same form as the complete integer solution [with integer parameters]? Clearly, $n=2$ does, and $n=3$ does not — see the famous Lebesgue Three-Square Identity (e.g. on your site). – Kieren MacMillan Sep 08 '14 at 21:30
  • While (2) works, it does NOT find all rational points on the unit sphere S2. For example the point [2/15, 2/3, 11/15] is missing. – Randall Oct 18 '23 at 02:43
  • New link : https://web.archive.org/web/20230326021830/http://sites.google.com/site/tpiezas/ – Guruprasad Jan 20 '25 at 14:11