Consider the equation mod 4 to conclude that $a,b$ and $c$ must be even. Let $a=2a_1,b=2b_1,c=2c_1$. Thus we get the equation $a_1^2+b_1^2+c_1^2=5030503$. Now the RHS is seven mod eight but only quadratic residues modulo eight are $0,1$ and $4$. Therefore the equation has no solutions.