Infinite solutions for $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$

Question

Given $(\frac{n+1}{n})^a\cdot (\frac{m+1}{m})^b = 2$ where a, b, n, and m are all positive integers, are there infinitely many solutions $(a,b,n,m)$?

Answer 1

Let's try this:

$(n+1)^{a}(m+1)^{b}=2\cdot n^{a}m^{b}$

Obviously either $(n+1)$ or $(m+1)$ must be even but not both. Take $(n+1)=2^{j}\cdot r;(n+1)^{a}=2^{a\cdot j}r^{a}$

Now $n+1$ is relatively prime to $n$ so if $a\cdot j>1$ then $m=2^{k}s$ and we have

$a\cdot j-b\cdot k=1$

Thus $(a,b),(a,j),(j,k),(j,b)$ are all relatively prime.

When we ``reverse engineer'' this we find a problem. We are left with:

$r^{a}\cdot(2^{k}s+1)^{b}=(2^{j}r-1)^{a}(s)^{b}$

Thus every factor of $r^{a}$ is a factor of $s^{b}$ and vise versa; $r=s,a=b$ .

And the positive resolution of.

$a\cdot j-b\cdot k=1,a=b$ is

$a=b=1,j-k=1$

$\left(\frac{2^{j}}{2^{j}-1}\right)\left(\frac{2^{j-1}+1}{2^{j-1}}\right)=2$

$j=2$

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