TITLE: Number which is simultaneously sum of 2 and 3 squares
QUESTION [15 upvotes]: 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.

REPLY [30 votes]: There is a possibly famous identity:
$$\Large 10^2+11^2+12^2=13^2+14^2$$ which meets your criteria.
Here $m$ is $365$.