Question

Determinati numerele naturale a si n pentru care a² - 12 =n! (n!=1×2×3...×n) si (0!=1)​

Answer 1

$\displaystyle\it\\a^2-12=n!,~a,n\in\mathbb{N}.\\a^2-12=n! \Leftrightarrow a^2=n!+12.\\daca~n\geq 5 \implies n!=\mathcal{M}_{10},~de~unde~n!+12=\mathcal{M}+2,~dar~un~patrat\\perfect~nu~poate~fi~de~forma~\mathcal{M}_{10}+2,~deci~ne~raman~de~analizat\\cazurile~n\in\left\{0,1,2,3,4\right\}.\\se~analizeza~cele~5~cazuri~si~se~observa~solutia~unica~n=4,~de~unde~a=6.$