I need to prove that n^4 - 4n^2 is divisible by 3.
The induction hypnosis would be k^4 - 4k^2 is indeed divisible by 3, for k >= 0.
What I don't understand is after we expand out (k+1) -4(k+1)^2, why do we need to subtract (n^4-4n)?
I know that n^4 - 4n = 3t for some integer t (divisible by 3).
The answer is given somewhere in the middle.
Thanks!Hence (n+1)^4 - 4(n+1)2 is divisible by 3 if (n+1)^4 - 4(n+1)^2-(n^4 -4n^2)
is divisible by 3.