Question

Arătaţi că ecuatia x² - 2020y² = 2021 - 2019 nu are solutii in multimea numerelor întregi.​

Answer 1

$x^2-2020y^2=2021-2019x \Leftrightarrow x^2+2019x=2021+2020y^2,\\observam~ca~2021+2020y^2=\mathcal{M}_{10}+1.\\ne~propunem~sa~aratam~ca~x^2+2019x\neq \mathcal{M}_{10}+1.\\U(x^2)+U(2019x)=U(x^2)+U(9x).\\U(x^2) \in \left\{ 0,1,4,5,6,9\right\}.\\se~analizeaza~toate~cazurile~si~se~observa~ca~ultima~cifra~a~U(x^2)+U(9x)\\nu~poate~fi~1.\\prin~urmare,~ecuatia~nu~are~solutii~intregi.$