| A Note on the Exponential Diophantine Equation $(a^m-1)(b^n-1)=x^2$ |
| Received:July 10, 2010 Revised:November 20, 2010 |
| Key Words:
Pell's equation congruences.
|
| Fund Project:Supported by the National Natural Science Foundation of China (Grant No.10901002). |
|
| Hits: 3555 |
| Download times: 3416 |
| Abstract: |
| Let $a$ and $b$ be fixed positive integers. In this paper, using some elementary methods, we study the diophantine equation $(a^m-1)(b^n-1)=x^2$. For example, we prove that if $a\equiv 2\pmod 6$, $b\equiv 3\pmod{12}$, then $(a^n-1)(b^m-1)=x^2$ has no solutions in positive integers $n,m$ and $x$. |
| Citation: |
| DOI:10.3770/j.issn:1000-341X.2011.06.014 |
| View Full Text View/Add Comment |
