# Internet Resources Collection on Equal Sums of Like Powers

• The Prouhet, Tarry, Escott problem revisited
• Peter Borwein ( pborwein@cecm.sfu.ca ) and Colin Ingalls
• Enseign.Math. (2).40.(1994),no.1-2,3-27, MR95d:11038
• Abstract. The author gives a collection of known elementary result, and discusses the most interesting case n=k+1.Some open problems are present.
• Computing Minimal Equal Sums Of Like Powers
• Jean-Charles Meyrignac's distributed-computing project on equal sums of like powers and the place to look for the current status of the problem.
• New results in equal sums of like powers
• Randy L. Ekl's article in Math. Comp. 67 (1998), pp. 1309-1315.
• Abstract. Given a positive integer n, find two sets of integers a_1, ..., a_r and b_1, ..., b_r, with r as small as possible, such that sum (a_j)^k = sum (b_j)^k for k = 1, 2, ..., n. Conjecture: r=n+1 for all n.
• Pure product polynomials and the Prouhet-Tarry-Escott problem
• Roy Maltby
• Math. Comp. 66 (1997), pp. 1323-1340
• Abstract. An n-factor pure product is a polynomial which can be expressed as a product where each factor is of the form (1-x^a_i) for some natural numbers a_i. We define the norm of a polynomial to be the sum of the absolute values of the coefficients. It is known that every n-factor pure product has norm at least 2n. We describe three algorithms for determining the least norm an n-factor pure product can have. We report results of our computations using one of these algorithms which include the result that every n-factor pure product has norm strictly greater than 2n if n is 7, 9, 10, or 11.
• Equal sums of four seventh powers
• Randy L. Ekl
• Math. Comp. 65 (1996), pp. 1755-1756.
• Abstract. In this paper, the method used to find the smallest, nontrivial, positive integer solution of a_1^7+a_2^7+a_3^7+a_4^7=b_1^7+b_2^7+b_3^7+b_4^7 is discussed. The solution is 149^7+123^7+14^7+10^7=146^7+129^7+90^7+15^7. Factors enabling this discovery are advances in computing power, available workstation memory, and the appropriate choice of optimized algorithms.
• Massively Parallel Number Theory
• Bob Scher ( bscher@peerlogic.com ) and Ed Seidl
• Abstract. Using a week of computer time on a 72 node Intel Paragon, we found the first new solution to the fifth order equation since 1966:
• 141325 + 2205 - 140685 - 62375 - 50275 = 0
• Euler's Sum of Powers Conjecture
• Eric W. Weisstein ( eww6n@carina.astro.virginia.edu )
• Abstract. Euler conjectured that at least n nth Powers are required for n>2 to provide a sum that is itself an nth Power. The conjecture was disproved by Leon J. Lander and Thomas R. Parkin in 1966 with the counterexample
• 1445 = 275 + 845 + 1105 + 1335
• Diophantine Equation
• Eric W. Weisstein ( eww6n@carina.astro.virginia.edu )
• Abstract. An equations in which only Integer solutions are allowed. Hilbert's 10th Problem asked if a technique for solving a general Diophantine existed. A general method exists for first degree Diophantine equation. However, the impossibility of obtaining a general solution was proven by Julia Robinson and Martin Davis in 1970. No general method is known for quadratic or higher Diophantine equations. J. P. Jones (1982) proved that no Algorithms can exist to determine if an arbitrary Diophantine equation in 9 variables has solutions. Ogilvy and Anderson (1988) give a number of Diophantine equations with known and unknown solutions.
• A^3+B^3+C^3=D^3
• Noam D. Elkies ( elkies@math.harvard.edu )
• Abstract. This is a famous Diophantine problem, to which Dickson's _History of the Theory of Numbers, Vol. II_ devotes many pages. Yes, there are plenty of examples, also of A^3+B^3=C^3+D^3 (which is mathematically the same thing if you change a sign) -- most famously 1729 = 12^3 + 1^3 = 10^3 + 9^3 -- and even known formulas for finding all solutions. Here is one which I worked out last year.
• Sums of square roots ( An application of the Prouhet-Tarry-Escott problem )
• David Eppstein ( eppstein@euclid.ics.uci.edu )
• Abstract. A major bottleneck in proving NP-completeness for geometric problems is a mismatch between the real-number and Turing machine models of computation: one is good for geometric algorithms but bad for reductions, and the other vice versa. Specifically, it is not known on Turing machines how to quickly compare a sum of distances (square roots of integers) with an integer or other similar sums, so even (decision versions of) easy problems such as the minimum spanning tree are not known to be in NP. Joe O'Rourke discusses an approach to this problem based on bounding the smallest difference between two such sums, so that one could know how precise an approximation to compute.
• Tarry-Escott Problem
• Equal Sums of 4th Powers
• Mathematics: Miscellanea
• Mersenne: extended Euler's conjecture
• Wide Area Distributed Computation: The Prouhet-Tarry-Escott Problem
• The Theorem of EL'Kahhar
• Diophantine's solutions when power n = no of terms
• a1^n + a2^n +...............+ aN^n=b^n , Can Integral solution be found for such equation?
• Whole number solutions?
• Does one need n number of terms to find whole number solutions to such an equation? For example, are there whole number solutions to: a^4 + b^4 + c^4 + d^4 = u^4 ??
• x^3+y^3=z^3+w^3
• Would it be difficult to write a program to develop a list of these numbers?
• Where to look for latest work re. proof of a^n + b^n + c^n = d^n (0 < n < ?)
• Given a^n + b^n + c^n = d^n where a, b, c, d, n are all positive integers.
• 1) Prove or disprove an upper bound for n.
2) If a solution exists for a particular n, then do solutions exist for all integers m where 0 < m < n?

Last revised July 14, 1999.