A sextic counterexample to Euler's sum of powers conjecture

Definitions and mathematical statement

The Diophantine existence problem is to find positive integers $x_1,\ldots,x_6$ satisfying \[ x_1^6+x_2^6+x_3^6+x_4^6+x_5^6=x_6^6, \] or to prove that there is no such tuple. Every variable is strictly positive, so no zero term can be used to reduce the number of summands. The five integers on the left need not be distinct. Permuting them does not change the equation, and one may impose $x_1\le\cdots\le x_5<x_6$ when searching.

Because the equation is homogeneous, multiplying all entries by the same positive integer preserves it. Existence is therefore equivalent to existence of a primitive tuple, meaning $\gcd(x_1,\ldots,x_6)=1$; divide by the common divisor to obtain one. It is also equivalent to the existence of positive rational numbers $y_1,\ldots,y_5$ with $\sum_i y_i^6=1$, by normalizing $x_6$ and then clearing denominators in the reverse direction. The question fixes both the exponent six and the number five of summands. Examples with a different exponent, an extra summand, or some zero coordinates do not satisfy this target. A positive answer can be certified by direct exact integer arithmetic.

Short English statement

Can a positive sixth power equal the sum of exactly five positive sixth powers, with repeated summands allowed?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3388 (OPG-508), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, A sextic counterexample to Euler's sum of powers conjecture. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/a_sextic_counterexample_to_eulers_sum_of_powers_conjecture.