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There is a set $Q$ which contains all integral values that can be represented by $$a^2 + b^2 + 4ab$$, where $a$ and $b$ are also integers. If some integers $x$ and $y$ exist in this set, prove that $xy$ does too.

I really have no idea how I can go about solving this. I tried simple multiplication of the two assuming one to be $(a^2 + 4ab + b^2)$ and other as $(c^2 + 4cd + d^2)$ but ultimately it leads to a long equation I can make no tail of :/

Any help whatsoever would be greatly appreciated

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I have updated your post to LaTeX. Please see that the updates are correct. –  gekkostate Jan 20 at 12:48
    
@hardmath Fixed! Thanks for the catch! –  gekkostate Jan 20 at 13:01

2 Answers 2

up vote 6 down vote accepted

Since $a^2+b^2+4ab=(a+2b)^2-3b^2$, your numbers are exactly the numbers of the form $x^2-3y^2$. Now $x^2-3y^2$ is the norm of the algrebraic number $x+y\sqrt{3}$, so you have the identity

$$ (x^2-3y^2)(u^2-3v^2)=(xu+3yv)^2-3(xv+yu)^2 $$

(multiplicativity of norms).

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Thank you so much, now I can finally sleep with this homework done. –  skatter Jan 20 at 12:54
3  
To make the resulting identity explicit in terms of $a, b, c, d$, if $f(x,y) = x^2 + 4xy + y^2$, then $$f(ac-bd,ad+4bd+bc) = f(a,b) f(c,d).$$ –  heropup Jan 20 at 13:13

Generalization:

Using Brahmagupta-Fibonacci Identity,

$$(a^2+nb^2)(c^2+nd^2)=(ac\pm nbd)^2+n(ad\mp bc)^2$$

$$n=-m\implies (a^2-mb^2)(c^2-md^2)=(ac\mp mbd)^2-m(ad\mp bc)^2$$

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