<s>[INST] Number which is simultaneously sum of 2 and 3 squares Is there positive integer $m$ such $m=x_1^2+x_2^2$ and $m=y_1^2+y_2^2+y_3^2$ where $x_i, y_j$ are nonzero integers. I have tried by hand for the ten natural numbers but I was not able to find such $m$.
Would be thankful for help.
 [/INST] Alternatively you could use a formula to generate examples; for example
$$m=\left(a^2+b^2+c^2\right)^2+\left(a^2-c^2\right)^2=\left(a^2+c^2\right)^2+\left(2cb\right)^2+\left(a^2+b^2-c^2\right)^2$$
originating from the identity
$$\left(a^2+b^2+c^2\right)^2-\left(a^2+b^2-c^2\right)^2=\left(2ca\right)^2+\left(2cb\right)^2$$
and using $$\left(a^2+c^2\right)^2-\left(a^2-c^2\right)^2=\left(2ca\right)^2$$
to substitute for $\left(2ca\right)^2$
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