Snail, I hope you are just trying to tease me.

Not at all. Just flexing my creaking mental muscles on an interesting problem. I thought that was what this thread was about. If you didn't want your friends "proof" discussed, why did you post it?

Hint: if the inequality holds for some n (> 1), it will also hold for n+1...

Yes, the assertion is true for all values of n greater than or equal to 5. No problem.

n can in fact be either 1, or 2, or 3, or 4, by that argument. This leaves us with checking all values of x from 1 to 99 - a "finite" task.

That falls short of your friends claim he had a proof of MtheGM's conjecture that 1 and 81 were the only numbers with this property. It does not rule out the possibility that there could be 3 and 4 digit numbers that share the porperty. OK, there aren't but that isn't the point.

Have you got back to your friend about this?

(All right, I might be teasing a bit.)