$k = n^3 + 1$ and $n>2$
$n^3 + 1 = (n + 1)(n^2 - n + 1)$
Prove that $k$ has at least two factors other than itself and $1$.
Can someone help me with this proof but not with induction.
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You already have the two factors, $n+1$ and $n^2-n+1$. If you show one fact is not $1$ and not $n^3+1$, it would immediately gollow that bith are. Show $1<n+1<n^3+1$ for $n>1$. – Bobby Ocean Apr 16 '22 at 17:31
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You also need to show that those two factors are distinct. – Captain Lama Apr 16 '22 at 17:32
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$n+1 = n^2 - n + 1 \iff n^2 - 2n = 0.$ It is given that $n > 2.$ – user2661923 Apr 16 '22 at 17:42