Jagan Eedula

13- March- 2008

Abstract:

In this paper we develop a method for determining Tarry Escott

Numbers and Ideal solutions for Degree K=3, K=5. We will apply it to

give Tarry Escott numbers to verify. The proof of this result is

completely elementary.

Introduction:

The Prouhet-Tarry-Escott problem can be stated as:

Given a positive integer n, find two sets of integer solutions { a1,

a2, ... , am } and { b1, b2, ... , bm } such that the integers in

each set have the same sum, the same sum of squares, cubes etc., up

to and including the same sum of nth powers, i.e., we are to find

solutions in integers of the system of equations

a1k + a2k + ... + amk = b1k + b2k + ... + bmk ( k = 1, 2, ...,

n )

Solutions of this system will be denoted here by the notation

{ a1 , a2 , ... , am } = {b1 , b2 , ... , bm } ( k = 1, 2, ...,

n )

The Method presented in this paper is Unconventional way of forming

the numbers. we have to write the numbers in matrix form, After that

draw a square or hexagon by combining the numbers in the matrix .

Then generate the numbers under each line by following the directions.

Method 1: Square formation method to get the Degree 3 number set.

Write the numbers in matrix format and form the numbers by method

explained in below example.

First set formation:

2 4

6 8

Combine the numbers in clock wise direction to form a single tarry

escott number like 2, 4 will be combined and formed into 24 (i.e.

2X10+4). Apply same for all arrows then below set will be generated.

A= {24, 48, 86, 62 }

Second set formation:

Apply the same method in anti clock wise direction for above numbers

to get second set.

B={42,26,68,84}

Number sets A,B are the solution for Tarry Escott Problem Degree 3.

I.e 24+48+86+62=42+26+68+84

242+482+862+622=422+262+682+842

243+483+863+623=423+263+683+843.

We can form the matrix with n rows n column. Below is the method for

3 rows 3 columns.

11 12 13

14 15 16

17 18 19

{111213, 131619, 191817, 171411}={131211,111417,171819,191613}

Method 2: Hexagon formation method to get the Degree 5 number set.

Below is the example to generate the numbers for degree 5.

First set formation:

11 12 13

14 15 16

17 18 19

Form the numbers in clock wise direction by going in hexagon shape.

A= {1213, 1316, 1618, 1817, 1714, 1412 }

Second set formation:

Form the numbers in anti clock wise direction by going in hexagon

shape.

B= {1312, 1214, 1417, 1718, 1816, 1613}.

Number sets A, B are the solutions for Tarry Escott Problem Degree 5.

Preparing the ideal solutions:

For matrix a a-c

b b-c

By using the method 1 following ideal solution will be derived

{11a-c, 10b+a , 10a-11c+b,11b-10c}={10a+b,11b-c,11a-10c, 10b-11c+a}.

of course same like we can generate different ideal solutions using

method 1 and method 2

Rules to form the Matrix:

1) It can be any length but number of rows and number of

columns size should be same

2) the matrix can be extended like below

a a+x a+2x a+3x ..

b b+x b+2x b+3x

c c+x c+2x c+3x ..

. . . .

. . . .

. . . .

. . . .

Rules to draw Shape on numbers:

While drawing the arrows following things need to be observed.

1) Shape should be equal n-gons, n should be even

2) While forming the shape the line should draw to next row or next

column with respect to current vertices. But should not jump to other

than next columns or other than next rows in case of 2X2 size matrix.

If it is n x n matrix then line should be

Drawn from current vertices to (n-1)th row or column.

Finding other shapes and forming the ideal solutions more than

degree 5 is still open

References:

[1]http://euler.free.fr/eslp/TarryPrb.htm

[2]http://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html

[3] E.M. Wright, Prouhet's 1851 Solution of the Tarry-Escott Problem

of. 1910, M.A.A. Monthly, 66 (1959), 199-201. [24] E.M. Wright