a^2 + b^3 + c^4 + d^5...

[jb at brennen.net]

Seems to me that given your existing work, it's pretty easy to show
that all numbers > 291 can be expressed as a sum of different powers.

For the target number N, take the largest integer a such that 2^a <= N.
If N-2^a can be represented as a sum of different powers, or if N-2^a
is zero, you trivially have a representation of N.

If N-2^a cannot be represented as a sum of different powers, but if
N-2^(a-1) can be, that also leads to a representation of N; note that
the representation of N-2^(a-1) cannot include an exponent as high as
(a-1), because we already established that N-2^(a-1)-2^(a-1) == N-2^a
cannot be represented, and N-2^(a-1)-3^(a-1) will be negative assuming
that N > 15.

So in order for N to be unrepresentable as a sum of different powers,
we would need to have both N-2^a and N-2^(a-1) unrepresentable.
Assuming that N>=1024, it is impossible for both of these to be
<= 291, so once you have established that all numbers from 292 - 1023
are representable, you have an actual constructive algorithm to
represent all numbers >= 1024.


Any flaws in that argument, other than the minor bit of
hand-waving?

  Jack