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For how many pairs of positive integers $(n, \space m)$ with $n, \space m < 100$ are both of the polynomials $x^2 + mx + n$ and $x^2 + mx - n$ factorable over the integers?

I have found four solutions: $$x^2 + 10x + 24$$ $$x^2 + 20x + 96$$ $$x^2 + 13x + 30$$ $$x^2 + 17x + 60$$

but I found these by trial and error.

I do not see how to systematically answer this question. Any help?

Kara
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    You need $m^2+4n$ and $m^2-4n$ to both be square. – Gerry Myerson Dec 09 '13 at 06:02
  • The rather stupid bruteforce program here (http://ideone.com/wuyoZZ) spits all the pairs $(m,n)$. You have missed several ones, but I don't see any obvious pattern. Namely they are $$5,6$$ $$10,24$$ $$13,30$$ $$15,54$$ $$17,60$$ $$20,96$$ $$25,84$$ – chubakueno Dec 10 '13 at 04:03

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