The Prouhet-Tarry-Escott problem for Gaussian integers

Abstract

Abstract. Given natural numbers n and k, with n > k, the Prouhet-TarryEscott (pte) problem asks for distinct subsets of Z, say X = {x 1 , . . . , xn} and Y = {y 1 , . . . , yn}, such thatMany partial solutions to this problem were found in the late 19th century and early 20th century.When n = k − 1, we call a solution X = n−1 Y ideal. This is considered to be the most interesting case. Ideal solutions have been found using elementary methods, elliptic curves, and computational techniques. In 2007, Alpers and Tijdeman gave examples of solutions to the pte problem over the Gaussian integers. This paper extends the framework of the problem to this setting. We prove generalizations of results from the literature, and use this information along with computational techniques to find ideal solutions to the pte problem in the Gaussian integers.

The Prouhet Tarry Escott Problem: A Review

“…Two years later, Prugsapitak [51] determined the complete ideal solutions of the PTE problem of degree two over both Z[i], the ring of Gaussian integers, and F p [x], the ring of polynomials over a finite field F p where p is a prime. A different approach to that of Cayley [52,53] was adopted. Also, a discussion on obtaining ideal solutions over Z[i] from the integer ideal solutions of degree two and from the proper integer solutions of the two-dimensional PTE problem was performed.…”

“…The developed method could be used to get infinitely many solutions of degree two over Z [i]. Cayley [52,53] Cerný [54] presented a class of solutions to the PTE problem. Prouhet's solution was described as a special case of that class.…”

“…Does an ideal solution exist for the PTE problem of size beyond 13? Is it possible to extend the algorithm found by Cayley [52,53] to find the solutions of larger size? The algorithm used in Reference [52] was an extension of the algorithm for finding odd and even symmetric solutions to the PTE problem over Z, which was implemented in Reference [42].…”

Three-Tangle for High-Rank Mixed States

“…For three-qubit systems, some results have been presented [8,9,10,11]. An important quantity for three-qubit entanglement is the so called residual entanglement or three-tangle [12], which is a polynomial invariant for three-qubit states, the modulus of the hyperdeterminant [13,14].…”

“…where p 0 =0.7377, p 1 =0.9559, g I (p) is given by (11) and g II (p) by (13). And the corresponding optimal decomposition are (10), (8) and (12) respectively.…”