Question

Fie a ∈ Z aratati c a numarul n=a(a+1)(a+2)(a+3)+1 este patrat perfect

Answer 1

$\displaystyle n=a(a+1)(a+2)(a+3)+1= \\ \\ ~~~= \Big[ a(a+3)\Big] \Big[ (a+1)(a+2) \Big]+1= \\ \\ ~~~=(a^2+3a)(a^2+3a+2)+1. \\ \\ Notam~a^2+3a=t \in \mathbb{Z}. \\ \\ Atunci~n=t(t+2)+1=t^2+2t+1=(t+1)^2 \longleftarrow patrat~perfect.$