AFAIK generalized taxicab numbers A^n + B^n = C^n + D^n for n>=5 would violate the <https://en.wikipedia.org/wiki/Lander,_Parkin,_and_Selfridge_conjecture> ("Euler`s extended conjecture") of 1967 stating that in sums of like powers the number of summands should be greater than or equal to the exponent. A site with much information about such sums is http://euler.free.fr -Georg Am 10.10.2020 um 15:59 schrieb Dan Asimov:
1) What are the arguments for/against the existence of generalized taxicab numbers A^n + B^n = C^n + D^n ?
2) Is there a reason to believe that these should exist only for finitely many values of n ≥ 1 and positive A, B, C, D ???
3) And what if (for odd n) A, B, C, D are allowed to be negative???
—Dan
----- On Fri, Oct 9, 2020 at 12:06 PM Frank Stevenson < frankstevensonmobile@gmail.com> wrote:
As a programming exercise I just finished a search for a solution A⁵ + B⁵ = C⁵ + D⁵ for numbers up to 4.25e37 ( 2 ^ 125 ) but did not find anything. ... ... -----
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