Given an equation that I've been looking at of the form $$(2^n-1)(1+x_1+x_1^2+...+x_1^{2j_1})(1+x_2+x_2^2+...+x_2^{j_2})\cdot \cdot \cdot(1+x_m+x_m^2+...+x_m^{j_m})+1=2^n\cdot x_1^{j_1}\cdot x_2^{j_2}...x_m^{j_m}$$ I can find a solution where $x_i$ are of the form $2^k$ and I want to show that this is the only possible non negative integer solution, but showing that no odd non negative solution is possible is also useful.
My attempt There is a surjective mapping from all integer solutions to this equation to numbers of the form $2^k$.
Suppose that an odd non negative solution exists, $y_1,y_2,...,y_m$ satisfying the equation above.
Then I can map solutions $y_1,y_2...,y_m \longrightarrow $ some powers of $2$ which satsify the equation with the same number of terms as $y_i$ and same exponents. So as an example suppose a solution existed with odd natural numbers $y_i$ satisfying: $$7(1+y_1+y_1^2+y_1^3+y_1^4)(1+y_2+y_2^2+y_2^3+y_2^4+y_2^5+y_2^6)(1+y_3+y_3^2)+1=8*y_1^4\cdot y_2^6 \cdot y_3^2$$ and I know that a solution in non negative powers of 2 exists namely $(8,8^5,8^{35})$ One nice thing about the solutions in powers of 2 is that I remove the greatest term the solution still holds so if I took away $(1+y_3+y_3^2)$ the equation still holds using $(8,8^5)$. But I don't think that simply having a mapping implies that since the equation still holds for my powers of 2 that the equation holds for my odd y terms in odd integers i.e. : $$7(1+y_1+y_1^2+y_1^3+y_1^4)(1+y_2+y_2^2+y_2^3+y_2^4+y_2^5+y_2^6)+1=8*y_1^4\cdot y_2^6 $$ Because if I continue this I will get that $$7(1+y_1+y_1^2+y_1^3+y_1^4)+1=8*y_1^4$$ which holds for $y_1=8$ but wont hold for an odd positive integer. So I'm just kind of stuck in terms of knowing if I might be on the right track . Any guidance is appreciated