Classroom note: Almost-isosceles right-angled triangles

We provide an elementary method to show that there exist infinitely many right-angled triangles with integral sides in which the lengths of the two non-hypotenuse sides differ by 1. The method also enables us to construct all such right-angled triangles recursively. 1. Introduction There does not exist any isoceles right-angled triangle with integral sides. Does there exist a right-angled triangle with integral sides in which the lengths of the two non-hypotenuse sides differ by I? We shall call such a triangle an almost-isoceles right-angled (AIRA) triangle. For an AIRA-triangle, there exist positive integers x and y such that the lengths of the sides are x, x+ 1 and y respectively with x 2 +(x+1)2 = y2. We shall call the triple (x, x+l,y) an AIRA-triple. An immediate example is the triple (3,4,5) and another one is (20,21,29). We would need a etc .. Are there infinitely many AIRA-triples? If so, is there a way to find all such triples? The answer to both questions is "yes" 1 and one can reduce the problem to a Pell's equation (see [1], p.357) and show that there are infintely many AIRA-triples. In this note, we shall however use an elementary method to show that there are infinitely many AIRA-triples and that all such triples can be obtained recursively.