Can Someone solve
$$x(x+1)(x+2)...(x+2020)=1$$
analitically rather than numerically and show me all steps. Also maybe with some preconditions, like only natural solutions, so that it‘s adaptable to diophantine equations, etc.
Thank you all in advance!!!
Write the equation like $$(x+1) (x+2) (x+3) (x+4) (x+5)\ldots(x+k)=\frac{1}{x}$$ The intersection between the polynomial $y=(x+1) (x+2) (x+3) (x+4) (x+5)\ldots(x+k)$ and the hyperbola $y=1/x$ are exactly $k$ if $k>3$.
This happens because $P_k(x)=(x+1) (x+2) (x+3) (x+4) (x+5)\ldots(x+k)$ has $k$ zeros and in any interval $(j-1,j);\;j\in\mathbb{Z},j<0$ there is a minimum $\xi_j$ and a maximum $\zeta_{j-1}$ for $j=\ldots,-3,-2,-1$.
TO DO
We must prove that the local minima $\xi_j$ have negative values $P(\xi_j)<\frac{1}{\xi_j}$ for $|j|>3$ therefore the curve $y=P_j(x)$ intersects the hyperbola in $|j|$ points for $|j|>3$.
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