The Pythagorean Triples Whose Hypotenuse and the Sums of the Legs are Squares

K Raja Rama Gandhi,Reuven Tint

doi:10.18052/www.scipress.com/bsmass.7.46

Abstract

More than 370 years ago the famous French mathematician Pierre de Fermat proposed to solve the following problem: to find a Pythagorean triples whose hypotenuse and the sum of the legs were squares, which, despite its simplicity, has been very difficult. Problems associated with its solution involved many mathematicians such as (Leonhard Euler, Joseph-Louis Lagrange, Ljnggren, Wacław Sierpiński and etc.) But in the end it did not reach solution. In our article, solution communicated to obtain the equations giving the required values all elements of the Pythagorean triples in positive integers (natural) are co-prime integers, and provides a second solution of thisproblem (the values of x, y, z of 45 digits), and some consequences.

Highlights

In 1643, Fermat challenged Mersenne to find a Pythagorean triplet whose hypotenuse and sum of the legs were squares. (“very difficult to solve the following problem”, Diophantus of Alexandria, “Arithmetica”, Наука, 1974 p. 309) Fermat found the smallest such solution
The first person who show how to obtain first solution of the problem, that resulted by Fermat was the Wacław Franciszek Sierpiński. He solved the problem of obtaining all results that include coprime solutions. (“Pythagorean triangle”, Uchpedgiz, 1959, §12)
Comment: [ Ljunggren proved that the equation has only two solutions in positive integers: 1 and 13

Summary

Introduction

In 1643, Fermat challenged Mersenne to find a Pythagorean triplet whose hypotenuse and sum of the legs were squares. (“very difficult to solve the following problem”, Diophantus of Alexandria, “Arithmetica”, Наука, 1974 p. 309) Fermat found the smallest such solution. The first person who show how to obtain first solution of the problem, that resulted by Fermat was the Wacław Franciszek Sierpiński. He solved the problem of obtaining all results that include coprime solutions. He has not led the explicit equations for solutions to this problem.

Results

Conclusion