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 a₁, a₂, ..., a_m and b₁, b₂, ..., b_n of natural numbers:

such that:

We call such lists a₁, a₂, ..., a_m and b₁, b₂, ..., 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

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: