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Does anyone know who was the first to find solutions in integers for $n^2 + (n+1)^2 = k^2$ (almost isosceles Pythagorean triples)? I know that Euler provided formulas for infinitely many $n, k$ that satisfy the equation.

Who proved (if that's the case) that there are no other solutions?

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I don't have the exact answer to this question of yours, but I know of a source wherein this problem is discussed in detail. Here is the relevant bibliographical information of it:

W. Sierpinski, Pythagorean Triangles. The Scritpa Mathematica Studies (No. 9); Published by the Graduate School of Science of Yeshiva University, NY, 1962.

The resolution of the Diophantine equation in which you are interested is discussed in sections 4.6-4.8 (pp. 16-20) of this book.

I hope this helps.

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