AoPS - Problem

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Problem

Source: Baltic Way 1994

Tags: number theory proposed, number theory

WakeUp

22.12.2011 17:18

Show that for any integer $a\ge 5$ there exist integers $b$ and $c$ , $c\ge b\ge a$ , such that $a,b,c$ are the lengths of the sides of a right-angled triangle.

WakeUp

22.12.2011 17:22

If $a=2k$ is even then note that $(2k)^2+(k^2-1)^2=(k^2+1)^2$ , and clearly $c\ge b\ge a$ . If $a=2k+1$ then we have $(2k+1)^2+(2k(k+1))^2=(2k^2+2k^2+1)^2$ and again $c\ge b \ge a$ is apparent. These have been worked out using the formula for pythagorean triples.