Collection on Equal Sums of Like Powers
Prouhet, Tarry, Escott problem revisited
Borwein ( firstname.lastname@example.org
) 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.
results in equal sums of like powers
- Randy L. Ekl's article in Math. Comp. 67 (1998), pp. 1309-1315.
Tarry-Escott multigrades problem
- 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.
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.
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.
Parallel Number Theory
Scher ( email@example.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
Sum of Powers Conjecture
W. Weisstein ( firstname.lastname@example.org
- 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
W. Weisstein ( email@example.com
- 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
D. Elkies ( firstname.lastname@example.org
- 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.
of square roots ( An application of the Prouhet-Tarry-Escott
Eppstein ( email@example.com
- 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.
Sums of 4th Powers
extended Euler's conjecture
Area Distributed Computation: The Prouhet-Tarry-Escott Problem
Theorem of EL'Kahhar
solutions when power n = no of terms
- a1^n + a2^n +...............+ aN^n=b^n , Can Integral solution be found
for such equation?
- 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 ??
- Would it be difficult to write a program to develop a list of these
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.
Copyright 1997-1999, Chen Shuwen