Prove that (n + 1)3 ≠ n3 + (n - 1)3 for any positive integer n.
Solution
If we have equality, then n3 = 6n2 + 2. But if n ≤ 6, then n3 ≤ 6n2 < 6n2 + 2. If n ≥ 7, then n3 ≥ 7n2 > 6n2 + 2. Contradiction.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03