An Elementary Proof of a Result Ma and Chen

Qing Han,Pingzhi Yuan

doi:10.11648/j.ajmcm.20200502.12

Abstract

In 1956, Je_smanowicz conjectured that, for positive integers m and n with m > n; gcd(m; n) = 1 and m ≢ n (mod 2), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution (x; y; z) = (2; 2; 2). Recently, Ma and Chen proved the conjecture if 4 ☓mn and y ≥ 2. In this paper, we provide a proposition that, for positive integers m and n with m > n; gcd(m; n) = 1 and m2 + n2 ≡ 5 (mod 8), the exponential Diophantine equation (m2 -n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution x = y = z = 2 with 2j gcd(x; y). Then we present an elementary and simple proof of the result of Ma and Chen by using Jacobi’s symbols.

Highlights

Let a, b and c be positive integers satisfying a2 + b2 = c2
We provide a proposition that, for positive integers m and n with m > n, gcd(m, n) = 1 and m2 + n2 ≡ 5, the exponential Diophantine equation (m2 − n2)x + (2mn)y = (m2 + n2)z has only the positive integer solution x = y = z = 2 with 2| gcd(x, y)
It is wellknown that a primitive Pythagorean triple (a, b, c) can be parameterized by a = m2 − n2, b = 2mn, c = m2 + n2, where m and n are relatively prime positive integers with m > n and m ≡ n

Summary

Introduction

Le [6] applied the theory of linear forms in two logarithms to prove that if n = 3, m ≡ 2 (mod 4) and m > 6000, Conjecture 1.1 is true. Takakuwa [20] extended the result of Guo, Le [6] by proving that if n = 3, 7, 11, 15 and m ≡ 2 (mod 4), Conjecture 1.1 is true. For the proof of the above Proposition 1.1, Ma and Chen [11] used some complicated computations of Jacobi’s symbols and a known result of Miyazaki ([13] Theorem 1.5), which is based on deep results on generalized Fermat equations via sophisticated arguments in the theory of elliptic curves and. We present an elementary proof of Proposition 1.1 by using Jacobi’s symbols, the computations of Jacobi’s symbols are more involved here

Some Lemmas

Conclusion