A survey of The Prouhet-Tarry-Escott Problem and its Generalizations

This paper explores the Prouhet-Tarry-Escott problem (PTE), the Generalized PTE problem (GPTE), and the Fermat form of Generalized PTE problem (FPTE). The GPTE problem extends the PTE problem by allowing different sets of exponents, while the FPTE problem considers cases where the number of integers in the two sets differs by one. Multigrade chains are also investigated, involving multiple sets of integers satisfying the GPTE system. The study of PTE and GPTE problems is further extended from integers to trigonometric functions. Three novel generalizations of the Girard-Newton Identities are introduced to solve the PTE and GPTE problems: the first extends the domain of exponents to all integers; the second further generalizes to a broader form; and the third focuses on odd integer exponents. The constant $C$ in the PTE and GPTE problems is investigated, and a novel approach is proposed by introducing the normalized GPTE problem with six conjectures to determine bounds for each variable. This enhances the efficiency of computer searches for ideal non-negative integer solutions that satisfy $\sum_{i=1}^{n+1} a_i^k = \sum_{i=1}^{n+1} b_i^k$ for $k = k_1, k_2, \dots, k_n$. A general process of computer searches for GPTE solutions is discussed, and reference code for the search program is provided. Several parametric solutions for GPTE are presented, along with parametric methods for ideal prime solutions. Six open problems related to the PTE and GPTE problems are proposed, and three approaches to address these problems are suggested. In the appendix, an overview of the research status is provided for the 296 types of GPTE and FPTE that have ideal solutions.