--- 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:

? for(a=0,99,print(kronecker(3*a+2,4*a+3)))

...prints out an endless stream of -1, which gives strong

evidence that 3*a+2 can never be a quadratic residue

modulo 4*a+3.