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You will see here:

Bill Dubuque's Slick $\sqrt{3}$ irrationality proof

What is the trick with modulus for proving irrationality?

What about $\sqrt{2}$

Can you prove this is irrational by that trick?

Community
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Amad27
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1 Answers1

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Assume $\sqrt{2} = \dfrac{m}{n} \to m^2 = 2n^2$. If taking $\pmod 4$, you see that the left side is $1 \pmod 4$ while the right side is $2 \pmod 4$ for odd $m,n$. For even $m,n$, you can use descending trick to get a contradiction.

DeepSea
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  • Varying this slightly, we can completely about the descending trick. By assuming $m$ and $n$ are coprime them $m$ even implies $n$ odd. Then following your argument, if $m$ is odd then LHS is 1 mod 4 and RHS is 2 or 0 mod 4. If $m$ is even then LHS is 0 mod 4 and RHS is 2 mod 4 – Robert Chamberlain Feb 28 '15 at 11:26