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History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.
2
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sums of rational squares
This is an answer to the second question, "what is the simplest direct proof?"
Let m be an integer satisfyng $\frac{a^{2}}{b^{2}}+\frac{c^{2}}{d^{2}}=m$. $a,b,c,d \in \mathbb{Z}$, gcd(a,b)=1, gcd(c,d …