Diophantus方程y^2+D^n=x^3的解法

Solution of Diophantine Equation y^2+ D^n= x^3

本文研究了Diophantus方程 y^2+D^n=x^3,y>0,(D,y)=1的解法,还给出了当D=P是素数,5≤p≤17时方程满足p(?)y,2(?)x的全部正整数解,及方程y^2+11~n=x^3的全部正整数解.

This paper deals solution of diophantine equation y2+Dn=x3, y>0, (D, y)= l (1) where D is a positive iteger given, x, y and n are variabies. Then when D =p is a prime and 5≤p ≤ l7, we obtain all postitive integer solutions of equation (1) with p y, 2 x. Moreover, we prove that all positive integer solutions of the diophantine equation y2+ 1ln=x3 are (x, y, n) = (443. 1l2t, 9324·ll3t, 6t+3), (l5·1l2t, 58·ll3t, 6t+l), (5·1l2t, 2·ll3t, 6t+2), (3·ll2t, 4·1l3t, 6t+ l), wher t is non -negative integer.