Find integral solutions to the equation
$$2x^2+y^2=2\times(1007)^2+1$$
I tried:
I rewrote the equation as $2x^2+y^2=2028099$. I found that $y_{max}=1424$ and $y$ must be odd, so I set $y=1424-(2k+1)$, where $k_{max}=711$. However I don't know how to proceed further.
Please Help!
Thanks!
Looking at the equation modulo 2 shows that $y$ is odd. Then looking at it modulo 4 shows that $x$ is odd as well. (The square of an odd number is always $1\bmod4$.)
A brute force search shows that the solutions $(x,y)$ are (335,1343), (593,1151), (965,407) and (1007,1). I fail to see enough structure in the answer to be able to explain it.