Page 1 of 1
Numere compuse
Posted: Thu Mar 12, 2009 2:00 am
by Marius Mainea
Aratati ca pentru orice n natural , numarul \( a=5^{5^{n+1}}+5^{5^n}+1 \) nu poate fi prim.
Concursul ,,Gh.Lazar'',2005
Posted: Sat Mar 14, 2009 8:21 pm
by Marius Mainea
Indicatie: Noteaza
\( a=5^{5^n} \)
Posted: Sat Mar 14, 2009 9:02 pm
by Claudiu Mindrila
Cu notatia de mai sus, avem:
\( a^{5}+a+1=a^{5}+a^{2}+a+1-a^{2}=a^{2}\left(a^{3}-1\right)+\left(a^{2}+a+1\right)=a^{2}\left(a-1\right)\left(a^{2}+a+1\right)+\left(a^{2}+a+1\right)=\left(a^{2}+a+1\right)\left(a^{3}-a^{2}+1\right) \) si cum \( a^{2}+a+1>1 \) si \( a^{3}-a^{2}+1>1 \) rezulta cerinta problemei.