Sure, 334^3 + 438^3 = 495^3 + 1. On 12/12/12, Dan Asimov <dasimov@earthlink.net> wrote:
And OMG it was just noon here!!!!!!!!!!!!
Anyway, as Ramanujan is famous for pointing out to Hardy,
9^3 + 10^3 = 12^3 + 1
.
This is a counterexample to Fermat's Last Theorem . . . almost. It's off by one.
QUESTION: Are there other examples of integers x, y, z for which
x^3 + y^3 = z^3 + 1
???
(Does the restatement (x-1)(x^2 + x + 1) = (z-y)(z^2 + zy + y^2) help?)
*Positive* integers x, y, z ???
And how about for integer exponents > 3 ???
--Dan
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