Re: Never square
- --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
> Anyone know why 4ab+3(a+b)+2 is never a square?Rewrite this as:
N = (4*a+3) * b + (3*a+2)
In order for N to be a square, we must have (3*a+2) be a
quadratic residue modulo (4*a+3).
Note that (3*a+2) is equal to -1/4 (modulo 4*a+3).
[It is the modular solution to 4x+1 == 0]
-1/4 is a quadratic residue iff -1 is a quadratic residue.
And finally, -1 is never a quadratic residue modulo 4*a+3;
this is a basic result of quadratic reciprocity.
Or for you PARI/GP types who don't care about all of the theory:
...prints out an endless stream of -1, which gives strong
evidence that 3*a+2 can never be a quadratic residue