Let $\mathbb N=\{0,1,2,\ldots\}$. By the Gauss-Legendre theorem on sums of three squares, any $n\in\mathbb N$ with $n\equiv1,2\pmod4$ can be written as $x^2+y^2+z^2$ with $x,y,z\in\mathbb N$. Clearly, one of $x,y,z$ has the same parity with $n$.
On July 25, 2009, I formulated the following conjecture.
Conjecture. Let $$N_1=14617,\ N_2=15618\ \ \text{and}\ \ N_3=25582.$$ Then, for each $i\in\{1,2,3\}$, any integer $n>N_i$ with $n\equiv1,2\pmod4$ can be written as$x_1^2+x_2^2+x_3^2$ with $x_1,x_2,x_3\in\mathbb N$ and $x_1\le x_2\le x_3$ such that $x_i\equiv n\pmod2$.
Any ideas towards the solution of this conjecture?
