The Prouhet-Tarry-Escott Problem (PTE Problem) is an old unsolved problem in number theory. Extensive accounts on it are available in [27] or [15].
In its most general setting the PTE Problem asks for two multisets of integers
(or, equivalently, of rational numbers)
and
such that
Solutions to the PTE Problem are known only for , and in all these cases (except s=9) it is known that there are infinitely many non-equivalent solutions.
Due to its large underdeterminacy as a polynomial equation system,
the PTE Problem is often considered in its more restrictive,
symmetric version. For s odd, the symmetric version takes the form
In the case s=9, only two solutions to the PTE Problem are known,
and they are both
odd symmetric solutions. They are, up to equivalence, generated by
This finding suggested forcibly that split relations of the type (13) are a frequent feature of odd symmetric PTE solutions, and led us to check the other odd symmetric solutions we knew for the presence of this phenomenon. Interestingly, of the more 265 odd symmetric solutions of size 7 that we computed in several months of CPU time, more than 86% have a relation analogous to (13), that is zi1+zi2+zi3=zj1+zj2+zj3+zj4=0. (Here, analogously, the i and j index sets are a disjoint partition of .)
While we do not yet have an explanation for this phenomenon, we have apparently discovered a new fact about the symmetric PTE solutions that had not been noted previously in the literature.