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Consider the set $S$ of all integers of the form $f(x,y) + f(z,w)$, where $x,y,z,w$ are integers, $$ f(x,y) = a x^2 + b x y + a^2 y^2,$$ and $a,b$ are integers with $$a > 1, \; \; 0 < b < 2 a^{3/2}, \; \; \gcd(a,b)=1. $$ How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force method, but to no avail. Perhaps someone could help?

Note: true if $f$ reduces to the principal form, also true for $f(x,y) = 3 x^2 + 8 x y + 9 y^2,$ result Noam Elkies. Also, by itself the set of numbers represented by the binary $f(x,y)$ is multiplicative, as $f$ is of order 3 in the class group. Indeed the real question is order three, I just felt this way saved time.

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Compare math.stackexchange.com/questions/103818/… –  Will Jagy Mar 1 '12 at 0:15

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