Percy the clever mathematical pig is thinking one day in his sty. He is chewing on a nutrituous diet of corn and soybeans. Suddenly, he has an idea! He thinks of a new kind of number, which he decides to call "Percy numbers", modestly enough.

He calls an n-digit positive integer is a Percy number if it is equal to the sum of the n^th powers of its digits. So for example, 371 is a Percy number, since 371 = 3³ + 7³ + 1³.

Prove that there are no Percy numbers with 1000 digits (i.e. there is no 1000-digit number that is equal to the sum of the 1000^th powers of its digits).

Can you find a smaller upper bound on the maximum number of digits a Percy number can have?

Can you find the biggest Percy number?