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Problem of the Day #340: (ir)Rationality of $\sqrt{2}$ February 22, 2012

Posted by Albert in : potd , trackback

Thanks to my dad for this problem.

While no positive integers $M$ and $N$ satisfy $M^2 = 2 \cdot N^2$, determine (with proof) whether there are infinitely many pairs of integers $M$ and $N$ such that $M^2 = 2 \cdot N^2 + 1$.
Thus, as $M, N$ increase, $\frac{M}{N}$ becomes a better approximation for $\sqrt{2}$.

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