Lander, Parkin, and Selfridge conjecture

The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem.

Background

Diophantine equations, such as the integer version of the equation a² + b² = c² that appears in the Pythagorean theorem, have been studied for their integer solution properties for centuries. Fermat's Last Theorem states that for powers greater than 2, the equation a^k + b^k = c^k has no solutions with three positive integers a, b, c. Extending the number of terms on either or both sides, and allowing for higher powers than 2, led to Leonhard Euler to propose in 1769 that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

In symbols, if $\sum _{i=1}^{n}a_{i}^{k}=b^{k}$ where n > 1 and $a_{1},a_{2},\dots ,a_{n},b$ are positive integers, then his conjecture was that n ≥ k.

In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for k = 5:^[1]

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵.

In subsequent years further counterexamples were found, including for k = 4. The latter disproved the more specific Euler quartic conjecture, namely that a⁴ + b⁴ + c⁴ = d⁴ has no positive integer solutions. In fact, the smallest solution, found in 1988, is

414560⁴ + 217519⁴ + 95800⁴ = 422481⁴.

Conjecture

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured^[2] that if $\sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}$ , where a_i ≠ b_j are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m+n ≥ k. The equal sum of like powers formula is often abbreviated as (k, m, n).

This implies in the special case of m = 1 that if

\sum _{i=1}^{n}a_{i}^{k}=b^{k}

(under the conditions given above) then n ≥ k−1.

For this special case of m = 1, known solutions satisfying the proposed constraint with n ≤ k, where terms are positive integers, hence giving a partition of a power into like powers, are:

k = 3

3³ + 4³ + 5³ = 6³.

k = 4

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴, (Roger Frye, 1988)

30⁴ + 120⁴ + 272⁴ + 315⁴ = 353⁴, (R. Norrie, 1911)

Fermat's Last Theorem states that in this case the conjecture is true.

k = 5

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵, (Lander, Parkin, 1966)

7⁵ + 43⁵ + 57⁵ + 80⁵ + 100⁵ = 107⁵, (Sastry, 1934, third smallest)

k = 6

(None known.)

k = 7

127⁷ + 258⁷ + 266⁷ + 413⁷ + 430⁷ + 439⁷ + 525⁷ = 568⁷, (M. Dodrill, 1999)

k = 8

90⁸ + 223⁸ + 478⁸ + 524⁸ + 748⁸ + 1088⁸ + 1190⁸ + 1324⁸ = 1409⁸, (Scott Chase, 2000)

k >= 9