I would prove this by considering the equation modulo 5. In this case, the 5y term will disappear because 5 is congruent to 0 mod 5. Thus, it suffices to show that
x^2 cannot be congruent to 3 mod 5 (since 243723 is congruent to 3 mod 5).
To do this, consider all the possibilities for x. We can have x=0,1,2,3,4 and so x^2=0,1,4,4,1 mod 5 respectively.
Thus, x^2 cannot be congruent to 3 mod 5 and so we are done.
Hope this helps!