Question

aratati ca 2^(2n+1) x 5^n +1 nu este patrat perfect

Answer 1

$\displaystyle\it\\2^{2n+1}\cdot5^{n+1},~nu~poate~fi~patrat~perfect~pentru~ca~\\orice~patrat~perfect~se~poate~scrie~ca~produs~de~doi\\factori~care~sunt~de~asemenea~patrate~perfecte.\\in~cazul~nostru~5^{n+1}~un~factor~al~produsului\\poate~fi~patrat~perfect~daca~n~este~un\\numar~impar,~dar~2^{2n+1},~nu~poate\\fi~niciodata~patrat~perfect~pentru\\n\in\mathbb{N},~2n+1~este~un~numar~\\impar,~asadar,~numarul~nu~poate~fi~patrat~perfect.$