Lander, Parkin, and Selfridge conjecture

In number theory, 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 Template:Mvar-th powers equals the sum of some other Template:Mvar-th powers, then the total number of terms in both sums combined must be at least Template:Mvar.

Background

Diophantine equations, such as the integer version of the equation Template:Math 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 Template:Math has no solutions in non-zero integers Template:Math. 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 Template:Mvar and Template:Mvar greater than 1, if the sum of Template:Mathth powers of positive integers is itself a Template:Mvarth power, then Template:Mvar is greater than or equal to Template:Mvar.

In symbols, if $\sum _{i=1}^n{a}_i^k=b^k$ where Template:Math and $a_1,a_2,\dots ,a_n,b$ are positive integers, then his conjecture was that Template:Math.

In 1966, a counterexample to Euler's sum of powers conjecture was found by Leon J. Lander and Thomas R. Parkin for Template:Math:

Template:Math.

In subsequent years, further counterexamples were found, including for Template:Math. The latter disproved the more specific Euler quartic conjecture, namely that Template:Math has no positive integer solutions. In fact, the smallest solution, found in 1988, is

Template:Math.

Conjecture

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured that if $\sum _{i=1}^n{a}_i^k=\sum _{j=1}^m{b}_j^k,$ where Template:Math are positive integers for all Template:Math and Template:Math, then Template:Math. The equal sum of like powers formula is often abbreviated as Template:Math.

Small examples with $m=n=\frac{k}{2}$ (related to generalized taxicab number) include

Template:Math (known to Euler)

and

Template:Math (found by K. Subba Rao in 1934).

The conjecture implies in the special case of Template:Math that if $$\sum _{i=1}^n{a}_i^k=b^k$$ (under the conditions given above) then Template:Math.

An interpretation of Plato's number is a solution for Template:Math

For this special case of Template:Math, some of the known solutions satisfying the proposed constraint with Template:Math, where terms are positive integers, hence giving a partition of a power into like powers, are:

Template:Math

Template:Math

Template:Math

Template:Math (Roger Frye, 1988)

Template:Math (R. Norrie, 1911)

Fermat's Last Theorem implies that for Template:Math the conjecture is true.

Template:Math

Template:Math (Lander, Parkin, 1966)

Template:Math (Sastry, 1934, third smallest)

Template:Math

(None known. As of 2002, there are no solutions whose final term is Template:Math. )

Template:Math

Template:Math (M. Dodrill, 1999)

Template:Math

Template:Math (Scott Chase, 2000)

Template:Math

(None known.)

Current status

It is not known if the conjecture is true, or if nontrivial solutions exist that would be counterexamples, such as Template:Math for Template:Math.

See also

• Beal's conjecture
• Fermat–Catalan conjecture
• Jacobi–Madden equation
• List of unsolved problems in mathematics
• Experimental mathematics (counterexamples to Euler's sum of powers conjecture, especially smallest solution for k = 4)
• Prouhet–Tarry–Escott problem
• Pythagorean quadruple
• Sums of powers, a list of related conjectures and theorems

References

• Template:Cite book

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
• Template:MathWorld
• Template:MathWorld
• Template:MathWorld
• Template:MathWorld
• Template:MathWorld
• Template:MathWorld
• Template: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