PROJECT DETAILS

Euler conjectured in 1772 that:
If a sum of n positive kth powers equals one kth power, then n >= k

Nowadays, we know that this conjecture is false, as the following counterexamples were discovered:

1445 = 1335+1105+845+275 (first discovery by Lander, Parkin and Selfridge in 1966 !)
206156734 = 26824404+153656394+187967604 (second discovery by Noam Elkies in 1986)
4224814 = 4145604+2175194+958004 (smallest terms by Roger Frye in 1988)
9668+5398+818 = 9548+7258+4818+3108+1588 (Scott I. Chase in 2000)

Given a triplet (k,m,n) of natural numbers with:

1 <= m <= n

We are looking for two lists a1, a2, ..., am and b1, b2, ..., bn of natural numbers:

a1 >= a2 >= ... >= am
b1 >= b2>= ... >= bn
with the following properties:
- a1 > 1 and a1 <> b1
- no integer > 1 divides all the numbers a1, a2, ..., am, b1, b2, ..., bn.

such that:

a1k+ a2k+ ... + amk = b1k+ b2k+ ... + bnk

We call such lists a1, a2, ..., am and b1, b2, ..., bn a solution to the triplet (k,m,n).

Now, let us denote with f(k,m) the function that returns the minimum number of terms n for given k and m.

We have the following property:

a1k+ a2k+ ... + amk+ 1k*c = b1k+ b2k+ ... + bnk+ 1k*c
thus:
f(k,m+c) <= f(k,m)+c

Our goal is to compute f(k, m).

Mathematicians worked on the problem and found some identities , but there is no known method which can compute f(k, m) for k > 4, so our current approach is brute-force decomposition.
Due to the size of this problem, we will restrict our search to k <= 32.

Computation gives us the currently known lower bounds of f(k,m) depending on k<=32 (equations for powers above 32 need too much terms for the moment) :

For m=1:
 k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 n 2 2 3 3 4 7 7 8 11 13 16 24 30 37 32
 k 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 n 70 50 60 70 65 75 95 105 124 137 155 163 204 173 191 317 230

For m+n:
 k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 m+n 3 3 4 4 5 6 8 8 10 12 14 14 17 19 23
 k 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 m+n 23 28 30 33 37 40 44 46 51 55 54 63 66 69 72 77 84

LEGEND:

 power (k) number of right terms (n), best lower bound currently known number of right terms (n), best lower bound conjectured number of right terms (n), best lower bound proved

Lander, Parkin and Selfridge conjectured in 1966 that:

(k,m,n) is never solvable if m+n < k

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