Pythagorean triples with consecutive legs

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The Pythagorean triple $3,4,5$ has consecutive legs ($4=3+1$). The next Pythagorean triple with consecutive integer legs has legs $20$ and $21$. Check that they really form a Pythagorean triple and compute its hypotenuse.

Solution

By the Pythagorean theorem the hypotenuse $c$ satisfies $c^2=20^2+21^2$: $c^2 = 400+441 = 841 = 29^2,$ so $c=\boxed{29}$. The triple $20,21,29$ is Pythagorean and has consecutive legs, as required.