| 摘要: | 我們考慮特別的整係數方程式去尋找整數解或有理數解。Ratat和Goormaghtigh觀察出當x,y,m,n為正整數時，(x,y,m,n)=(5,2,3,5)和(90,2,3,13)是方程式 (x^m-1)/(x-1)=(y^n-1)/(y-1) 的解。因此，猜想此方程式只有這兩組解。現在，我們集中焦點在m=3。此時方程式有兩組已知的解。除了那兩組解之外的解就稱為例外解。這篇論文，主要是考慮當n=4時，此方程式沒有例外解。 We consider special Diophantine equations with integral coefficient and seek integral or rational solutions. Ratat[1] and Goormaghtigh [2] observed that 31=(2^5-1)/(2-1)=(5^3-1)/(5-1) and 8191=(2^13-1)/(2-1)=(90^3-1)/(90-1) are solutions of the Diophantine equation (x^m-1)/(x-1)=(y^n-1)/(y-1) ; x > 1; y > 1; n > m > 2.....(1) Now, we will focus our attention on the equation (x^3-1)/(x-1)=(y^n-1)/(y-1) ; n > 2; x > 1; y > 1 with x > y.....(2) Equation (2) has two known solutions (x, y, n) = (5, 2, 5), (90, 2, 13). Any other solution (x, y, n) of (2) will be called exceptional. In this paper, we show that this equation (2) has no exceptional solution when n = 4. |
