LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many que...LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.展开更多
文摘LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.
