Hall's Conjecture
About successive nth powers. i.e.
a0^n = a1^n + a2^n + a3^n + a4^n + ... + am^n [aj – a(j+1) = 1]
Proof:
m = 1 clearly not possible in integers
m = 2, n = 1, 3 = 2 + 1
m = 2, n = 2, 5^2 = 4^2 + 3^2
m = 3, n = 3, 6^3 = 5^3 + 4^3 + 3^3
m = 4, n = 4, not poss. [7 < a0 < 8]
m = 4k+1, not poss. [mod 2]
m > 5, not poss. [by extension of the immediately above argument to include arbitrary numbers of zeros]