Question

Demonstați că printre numerele 14, 144, 1444, 14444, ... sunt exact două pătrate perfecte.

Answer 1

$Observam~ca~144=12^2~si~1444=38^2.~Vom~demonstra~ca~acestea \\ \\ sunt~singurele~patrate~perfecte~din~acel~sir. \\ \\ Evident,~14~nu~este~patrat~perfect,~deci~ramane~de~demonstrat~ \\ \\ nu~exista~patrate~perfecte~de~forma~1\underbrace{4...4}_{\mbox{n}},~unde~n \in \mathbb{N},~n \geq 5.$

$Pentru~aceasta~ne~vom~folosi~de~resturile~modulo~16~ale~unui \\ \\ patrat~perfect.~Acestea~sunt~0,~1,~4~si~9~(vezi~tabelul~de~mai~jos) \\ \\ Insa~restul~impartirii~unui~numar~natural~la~impartirea~cu~16~ \\ \\ este~egal~cu~restul~impartirii~numarului~format~de~ultimele~4 \\ \\ cifre~ale~sale~cu~4.$

$Deci~1 \underbrace{4...4}_{\mbox{n}} \equiv 4444 \pmod{16} \equiv 12 \pmod {16}. ~n \geq 4)\\ \\ ~Insa~am~stabilit ~ anterior~ca~k^2 \equiv0,1,4,9 \pmod{16} ~\forall~k \in \mathbb{N}. \\ \\ Prin~urmare~niciun~numar~de~forma~1\underbrace{4...4}_{\mbox{n}}~(n \geq 4)~nu~este \\ \\ patrat~perfect.$