Problem Statement

There is a conjecture in mathematics that held for over two hundred years before it was disproved by the finding of a counterexample in 1966 by Lander and Parkin. This conjecture is called Euler's sum of powers conjecture and can be stated as such: In 1966, Leon J. Lander and Thomas R. Parkin used a brute-force search on a CDC 6600 computer restricting numbers to those less than 250. The task consists in writing a program to search for an integer solution of

x

0

5

x

1

5

x

2

5

x

3

5

=

y

5

{\displaystyle x_{0}^{5}+x_{1}^{5}+x_{2}^{5}+x_{3}^{5}=y^{5}}

where all

x

i

{\displaystyle x_{i}}

and

y

{\displaystyle y}

are distinct integers between 0 and 250 (exclusive). Show an answer here. Related tasks are:

Let's start with the solution:

(ql:quickload :alexandria)
(let ((fifth-powers (mapcar #'(lambda (x) (expt x 5))
                            (alexandria:iota 250))))
  (loop named outer for x0 from 1 to (length fifth-powers) do
    (loop for x1 from 1 below x0 do
      (loop for x2 from 1 below x1 do
        (loop for x3 from 1 below x2 do
          (let ((x-sum (+ (nth x0 fifth-powers)
                          (nth x1 fifth-powers)
                          (nth x2 fifth-powers)
                          (nth x3 fifth-powers))))
            (if (member x-sum fifth-powers)
                  (return-from outer (list x0 x1 x2 x3 (round (expt x-sum 0.2)))))))))))