WDS said "[Jeff D. Caldwell also was suggested to me, erroneously, that you could say the same about the form x^2-y^2 = (x-y)*(x+y). That does not quite work because x-y and x+y necessarily have the same parity, hence this form cannot represent integers congruent to 2 mod 4. But his idea does work if applied to x*y.]" I can't quite parse "Jeff D. Caldwell also was suggested to me, erroneously ...", but if you mean something I suggested to you, I suggested not x^2-y^2 but -x^2+xy. (ax^2+bxy+cy^2 with a=-1, b=1, c=0) (That's twice in two posts that you've misquoted me.) Integers 2 mod 4 certainly are represented in my form. Considering only naturals (which you stated in your conjecture #2 was your domain of interest), taking y from 1 to 12 and for each y, taking x from 1 to y - 1, gives: 1 2 2 3 4 3 4 6 6 4 5 8 9 8 5 6 10 12 12 10 6 7 12 15 16 15 12 7 8 14 18 20 20 18 14 8 9 16 21 24 25 24 21 16 9 10 18 24 28 30 30 28 24 18 10 11 20 27 32 35 36 35 32 27 20 11 The first and last entry in each row show that every natural is represented, including those that are 2 mod 4. The rows, if plotted, form nested parabolas that have points with simultaneously integer values of both x and y only when x is a factor of y (counting both 1 and y as factors of y). When plotted with the ranges shown above, the parabolic segments can be joined in a boustraphedonic path through the naturals and, given the x,y indices, their factors. For y a prime, the only intersection points with integer indices are x=1 and x=y. Perfect squares have integer x,y values at the tip of each respective parabola, where the row begins with an odd number. Thus, primes squared appear 3 times. All other integers generated by the form are less than 1. The values shown are repeated for negative x. The form produces all naturals in a bounded manner. Of course, my form no longer applies to your revised conjecture, which examines only positive definite and ternary or higher quadratic forms. I look forward to reviewing your proof in more detail. Jeff