A Diophantine equation

Number Theory Level 3

Are there infinitely many solutions to the equation $\large m^{2}-5n^{2}=1$ , where m and n are positive integers?

No Yes

1 solution

Freddie Hand
Feb 5, 2017

By trying out a few values, we find that $m=9$ and $n=4$ is a solution.

Now we can use induction.

If $(m,n)$ is a solution to $m^{2}-5n^{2}=1$ ,then $(m^{2}-5n^{2})^{2}=1=(m^{2}+5n^{2})^{2}-5(2mn)^{2}$ ,

Therefore, $(m^{2}+5n^{2},(2mn)^{2})$ is also a solution. Therefore, there are infinitely many solutions to the equation $m^{2}-5n^{2}=1$ .