Composite

[MM]

Prove that there exist infinitely many composite numbers of the form
[tex]\left(2^{n}+1\right)^{2}+4[/tex].

[broniran]

For all [tex]n\equiv 3(mod 4)\Rightarrow(2^n+1)^2+4\equiv 0(mod 5)[/tex].That is true because [tex]2^{4k+3}\equiv 8(mod 5)\Rightarrow 2^{4k+3}+1\equiv 9\equiv -1(mod 5)[/tex] which implies [tex](2^{4k+3}+1)^2\equiv 1(mod 5)[/tex]