Nowadays, we know that this conjecture is false, as the following counterexamples were discovered:
Given a triplet (k,m,n) of natural numbers with:
We are looking for two lists a_{1}, a_{2}, ..., a_{m} and b_{1}, b_{2}, ..., b_{n} of natural numbers:
such that:
We call such lists a_{1}, a_{2}, ..., a_{m} and b_{1}, b_{2}, ..., b_{n} 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:
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: