Lander, Parkin, and Selfridge conjecture

Last updated March 02, 2024

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. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.

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 in non-zero 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 nkth 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 = 144 ⁵.

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).

Small examples with $m=n={\frac {k}{2}}$ (related to generalized taxicab number) include $59^{4}+158^{4}=133^{4}+134^{4}$ (known to Euler) and $3^{6}+19^{6}+22^{6}=10^{6}+15^{6}+23^{6}$ (found by K. Subba Rao in 1934).

The conjecture 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, some of the 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:^[3]

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 for k = 4 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. As of 2002, there are no solutions whose final term is ≤ 730000.^[4] )

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

(None known.)

Current status

It is not known if the conjecture is true, or if nontrivial solutions exist that would be counterexamples, such as a^k + b^k = c^k + d^k for k ≥ 5. ^[5]^[6] Trivial solutions include certain cases with composite exponents k, e.g. k = 6, as for such k = p * q it is possible to have solutions (a^p)^q + (b^p)^q = (a^q)^p + (b^q)^p, for positive integers a and b.

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References

↑ L. J. Lander; T. R. Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72: 1079. doi: 10.1090/S0002-9904-1966-11654-3 .
↑ L. J. Lander; T. R. Parkin; J. L. Selfridge (1967). "A Survey of Equal Sums of Like Powers". Mathematics of Computation . 21 (99): 446–459. doi: 10.1090/S0025-5718-1967-0222008-0 . JSTOR 2003249.
↑ Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions" . Retrieved 17 July 2017.
↑ Giovanni Resta and Jean-Charles Meyrignac (2002). The Smallest Solutions to the Diophantine Equation $a^{6}+b^{6}+c^{6}+d^{6}+e^{6}=x^{6}+y^{6}$ , Mathematics of Computation, v. 72, p. 1054 (See further work section).
↑ A. Bremner; R. K. Guy (1998). "A Dozen Difficult Diophantine Dilemmas". American Mathematical Monthly . 95 (1): 31–36. doi:10.2307/2323442. JSTOR 2323442.
↑ T.D. Browning (2002). "Equal sums of two kth powers". Journal of Number Theory . 96 (2): 293–318.

Guy, Richard K. (2004). Unsolved Problems in Number Theory. Problem Books in Mathematics (3rd ed.). New York, NY: Springer-Verlag. D1. ISBN 0-387-20860-7. Zbl 1058.11001.

External links

EulerNet: Computing Minimal Equal Sums Of Like Powers
Jaroslaw Wroblewski Equal Sums of Like Powers
Tito Piezas III: A Collection of Algebraic Identities
Weisstein, Eric W. "Diophantine Equation--5th Powers". MathWorld .
Weisstein, Eric W. "Diophantine Equation--6th Powers". MathWorld .
Weisstein, Eric W. "Diophantine Equation--7th Powers". MathWorld .
Weisstein, Eric W. "Diophantine Equation--8th Powers". MathWorld .
Weisstein, Eric W. "Euler's Sum of Powers Conjecture". MathWorld .
Weisstein, Eric W. "Euler Quartic Conjecture". MathWorld .
Weisstein, Eric W. "Diophantine Equation--4th Powers". MathWorld .
Euler's Conjecture at library.thinkquest.org
A simple explanation of Euler's Conjecture at Maths Is Good For You!
Mathematicians Find New Solutions To An Ancient Puzzle
Ed Pegg Jr. Power Sums, Math Games

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[1] L. J. Lander; T. R. Parkin (1966). "Counterexample to Euler's conjecture on sums of like powers". Bull. Amer. Math. Soc. 72: 1079. doi: 10.1090/S0002-9904-1966-11654-3 .

[2] L. J. Lander; T. R. Parkin; J. L. Selfridge (1967). "A Survey of Equal Sums of Like Powers". Mathematics of Computation . 21 (99): 446–459. doi: 10.1090/S0025-5718-1967-0222008-0 . JSTOR 2003249.

[meyrignac-3] Quoted in Meyrignac, Jean-Charles (14 February 2001). "Computing Minimal Equal Sums Of Like Powers: Best Known Solutions" . Retrieved 17 July 2017.

[4] Giovanni Resta and Jean-Charles Meyrignac (2002). The Smallest Solutions to the Diophantine Equation $a^{6}+b^{6}+c^{6}+d^{6}+e^{6}=x^{6}+y^{6}$ , Mathematics of Computation, v. 72, p. 1054 (See further work section).

[5] A. Bremner; R. K. Guy (1998). "A Dozen Difficult Diophantine Dilemmas". American Mathematical Monthly . 95 (1): 31–36. doi:10.2307/2323442. JSTOR 2323442.

[6] T.D. Browning (2002). "Equal sums of two kth powers". Journal of Number Theory . 96 (2): 293–318.

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