丢番图方程的解

An Invitation to Modern Number Theory Pub Date : 2020-07-21 DOI:10.2307/j.ctv131bw89.9

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摘要

．本文研究了当1≤x, y, z≤2时，当质数p≥2时，Diophantine方程p x + (p + 1) y + (p + 2) z = m2的解。我们建立:(i)当p = 2且x = y = z = 1时，方程有唯一解。(ii)当p = 4n + 1且1≤x, y, z≤2时，方程无解。(iii)当p = 4 N + 3, x = y = z = 1时，方程有无穷多个解。(iv)当3≤p≤199,x = 1, y = z = 2时，方程只有一个解。(v)除上述情况外，1≤x, y, z≤2时，方程无解。

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Solutions to Diophantine Equations

. In this article, we investigate the solutions of the Diophantine equations p x + ( p + 1) y + ( p + 2) z = M 2 for primes p ≥ 2 when 1 ≤ x , y , z ≤ 2. We establish : (i) When p = 2 and x = y = z = 1, the equation has a unique solution. (ii) When p = 4 N + 1 and 1 ≤ x , y , z ≤ 2, the equations have no solutions. (iii) When p = 4 N + 3 and x = y = z = 1, the equation has infinitely many solutions. (iv) When 3 ≤ p ≤ 199 and x = 1, y = z = 2, the equation has exactly one solution. (v) In all other cases 1 ≤ x , y , z ≤ 2 which are not mentioned above, the equations have no solutions.

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An Invitation to Modern Number Theory

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