## Euler’s Sum Of Powers Conjecture

### April 17, 2015

Fermat’s Last Theorem, which dates to the seventeenth century states that there are no solutions in integers to the equation *x ^{n}* +

*y*=

^{n}*z*for

^{n}*n*> 2; the Theorem was finally proved a few years ago by Andres Wiles. In the eighteenth century, Euler conjectured that for any

*n*> 2, it would take at least

*n*terms of the form

*x*to sum to an

_{i}^{n}*n*th power. That conjecture held until the age of computers, in 1967, when Lander and Parkin found the counter-example 27

^{5}+ 84

^{5}+ 110

^{5}+ 133

^{5}= 144

^{5}.

Your task is to write a program that finds counter-examples to Euler’s Conjecture. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

In Python. I could not find anything clever.

A much faster method from the Lander and Parkin paper. The answer comes in a flash. I feel this can be improved, but this is already much faster.

@Paul: Please provide a citation to the paper.

@ programmingpraxis: the paper: Lander and Parkin

I made a straightforward implementation. I feel it can be speeded up more. I did not use the bit at the bottom of page 101 with the mod 30 stuff.

Another version in Python. It only produces “primitive” solutions, where the terms do not have a common factor. It produces the 2 tables from the Lander and Parkin paper. Also an example from Wikipedia for k=7 is found after a long time (about 40 minutes).

Somehow the superscript 5’s got left out of the 2 last terms, I found this in the html source: 133 = 144

@Marijn: Fixed. Thanks.