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An algebric characterization of right triangles (I like it)

Source: Romanian District Olympiad 2016, Grade IX, Problem 4

October 4, 2018
geometryalgebrapolynomialnumber theory

Problem Statement

Let a≥2 a\ge 2 be a natural number. Show that the following relations are equivalent:
(i) a \text{(i)} \ a is the hypothenuse of a right triangle whose sides are natural numbers. \text{(ii)}  there exists a natural number d d for which the polynoms X2−aX±d X^2-aX\pm d have integer roots.
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