In number theory, the Lander, Parkin, and Selfridge conjecture (the LPS conjecture) is an unsolved conjecture about Diophantine equations involving equal sums of like powers. It predicts that every nontrivial equality
x 1 k + x 2 k + ⋯ + x m k = y 1 k + y 2 k + ⋯ + y n k {\displaystyle x_{1}^{k}+x_{2}^{k}+\cdots +x_{m}^{k}=y_{1}^{k}+y_{2}^{k}+\cdots +y_{n}^{k}}
between sums of positive integer k {\displaystyle k} th powers must contain at least k {\displaystyle k} terms in total; in other words,
m + n ≥ k . {\displaystyle m+n\geq k.}
Here, “nontrivial” means that no term occurs on both sides after common terms have been cancelled. The statement arose from a question posed by Leon J. Lander, Thomas R. Parkin, and John Selfridge in their 1967 survey of equal sums of like powers. The conjecture remains open. In particular, no nontrivial positive-integer solution is known to
a 5 + b 5 = c 5 + d 5 . {\displaystyle a^{5}+b^{5}=c^{5}+d^{5}.}
Statement Let k ≥ 2 {\displaystyle k\geq 2} and let m , n ≥ 1 {\displaystyle m,n\geq 1} . Since the two sides may be interchanged, one may assume that m ≤ n {\displaystyle m\leq n} . Consider an equation
∑ i = 1 m x i k = ∑ j = 1 n y j k , {\displaystyle \sum _{i=1}^{m}x_{i}^{k}=\sum _{j=1}^{n}y_{j}^{k},}
where all the x i {\displaystyle x_{i}} and y j {\displaystyle y_{j}} are positive integers and
x i ≠ y j {\displaystyle x_{i}\neq y_{j}}
for every i {\displaystyle i} and j {\displaystyle j} . Repetitions within the same side of the equation are not excluded. The LPS conjecture states that such an equation can have a solution only if
m + n ≥ k . {\displaystyle m+n\geq k.}
An equation of this form is commonly described as having type ( k , m , n ) {\displaystyle (k,m,n)} . The original survey used the notation ( k . m . n ) {\displaystyle (k.m.n)} . A solution is called primitive if
gcd ( x 1 , … , x m , y 1 , … , y n ) = 1. {\displaystyle \gcd(x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n})=1.}
Every nonprimitive solution is obtained from a primitive one by multiplying all the bases by a common positive integer. In the one-sided case m = 1 {\displaystyle m=1} , the conjecture says that
a 1 k + a 2 k + ⋯ + a r k = b k ⟹ r ≥ k − 1. {\displaystyle a_{1}^{k}+a_{2}^{k}+\cdots +a_{r}^{k}=b^{k}\quad \Longrightarrow \quad r\geq k-1.}
This is weaker by one summand than Euler's sum of powers conjecture, which asserted that r ≥ k {\displaystyle r\geq k} .
Historical background Euler's sum of powers conjecture was intended as a generalization of Fermat's Last Theorem. It asserted that a k {\displaystyle k} th power could not be written as the sum of fewer than k {\displaystyle k} positive k {\displaystyle k} th powers. In 1966, Lander and Parkin disproved Euler's conjecture by finding
… excerpt ends here. Continue reading the full article.
