Explicit Examples of Optimal PTE Solutions for Degree 4 and the Concept of Absolute Universal Optimal Construction

📧 Contact: sasa.jov.ubl@gmail.com. This paper complements the impossibility results established in our earlier work, which proved that no universal construction can exist for ideal Prouhet–Tarry–Escott (PTE) solutions (those of minimal cardinality k + 1). There we identified k = 4 as the breaking point where any deterministic construction-chain terminates, precisely because the rigid algebraic constraints of the ideal regime forbid the recursive propagation required by universality. Here we demonstrate that universality can be restored by relaxing the size constraint: we present explicit families of degree-4 PTE solutions with set size n = 7 (two elements larger than the ideal size 5, yet nine elements smaller than the classical 24 = 16 of Prouhet’s 1851 construction). These solutions admit a deterministic one-directional propagation of informations, and they come in infinite parametrised families, parametrised either by the first or by the last element of the set B. 📘 Cite as: Saša Jovičević (2026), Explicit Examples of Optimal PTE Solutions for Degree 4 and the Concept of Absolute Universal Optimal Construction, Zenodo.