How do I prove there is no right triangle with integer legs and a hypotenuse of $1000\sqrt3$?
2026-05-06 03:11:39.1778037099
A right triangle: integer legs and a hypotenuse of $1000\sqrt3$
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If $1000\sqrt3$ was the hypotenuse of a right triangle with integer legs, we could write its square, 3 million, as the sum of two squares. However, this factors as $2^6\cdot3\cdot5^6$, the prime 3 of form $4n+3$ appearing an odd number (1) of times. Therefore 3 million is not a sum of two squares and the given hypotenuse cannot be that of an integer-legged right triangle.