Integer solution $xy=2^x-1$

Question

Find positive integer $x,y$ such that $$xy=2^x-1$$ I have made several attempts, but I don't know if this is totally correct: Notice that $x$ never divides $2 ^ x-1$.

Claim: There is no positive integer greater than $1$ such that $x$ divides $2^x-1$. We can assume that $\frac{2^x-1}{x}$ is an integer, and now we can consider the polynomial $$p(x)=\left(x-\frac{2^x-1}{x}\right)(x-1)x$$ After some algebraic manipulation we obtain $$x^3-x^2-x(2^x-1)+(2^x-1)=0$$ We know that one of the solutions is $0$ but we ignore this because $x$ is never equal to $0$. By Vieta's formula we have that the sum of other two solutions is $1$, but this is a contradiction because we supposed that $\frac{2^x-1}{x}\geq1$, because $x\geq1$. So that unique solution is $(1,1)$ Do you think it is correct? And if so, are there alternative solutions to this? Is it possible to find a way to do this by induction or infinite descent? Thanks

Your answer is wrong. You can't tell the degree of the quotient. There could be $x$ in it, and you can't rule that out. Thus the application of Vieta formula is incorrect. — Wang YeFei, Aug 13 '22 at 03:33
@Math Welcome to Math SE. As Wang YeFei's comment indicates, note that Vieta's formulas, with the roots, works with polynomials using constant coefficients, but with what you've got, the $x$ and constant value coefficients change with $x$. Instead, regarding your request for alternative solutions, consider the proposed duplicate questions below. — John Omielan, Aug 13 '22 at 03:34
Does this answer your question? Can an odd number $n$ divide $2^n-1$? - found using an Approach0 search. Another duplicate is Show that $n$ does not divide $2^n - 1$ where $n$ is an integer greater than $1$?, and there's the AoPS thread n| 2^{n} -1 is not possible for n>1. — John Omielan, Aug 13 '22 at 03:35
$p(x)$ is not a polynomial. Of $x$ is your given constant, then $p(x)$ is a constant. If $x$ is a variable, then the existence of $2^x$ makes it not a polynomial. — Thomas Andrews, Aug 13 '22 at 03:52

Integer solution $xy=2^x-1$

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