I'm trying to prove the following equation has no solutions, with the following conditions on a, b, c and d:
These 4 variables are integers. They can be negative, but each of them must have a unique absolute value. That is, if one of the variables is equal to x, then none of the others may be -x. Note that this rules out the trivial solution a = b = c = d = 0.
The equation is
3a^2 + a + 3b^2 + b = 3c^2 + c + 3d^2 + d
What techniques could I employ to show this has no solutions?
Both sides of the equation must also be equal to 2e^2, which means they must be equal to a number that is 2 times a perfect square.