Vitanuova for 2003 June 22 (entry 1)

Proof

On June 2 I presented a proof of a toy theorem which inverts Fermat's Last Theorem. Matthew Loran sent me a criticism of the proof, which I misunderstood, and then Andrew Cairns sent me a counterexample along with a similar criticism.

The proof is wrong. The problem is that I went from

n^y-z(n^a-y - n^b-y - n^c-y - ... - n^x-y - 1) = 1

to conclude that

n^a-y - n^b-y - n^c-y - ... - n^x-y = 1

where I should have concluded that

n^a-y - n^b-y - n^c-y - ... - n^x-y - 1 = 1

which is very different.

Andrew Cairns points out that some of the powers might be distinct and others might not. For example, we might have 128 = 64 + 32 + 32 (or, 729 = 243 + 243 + 81 + 81 + 81). What I think survives the error in my proof is the conclusion that a power of n (for n>2) cannot be decomposed into m distinct powers of n (or into m identical powers of n) if m does not equal n.

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