Marshall Hall’s conjecture

Conjecture. (Marshall Hall, Jr.). With the exception of n2 being a perfect sixth power, for any positive integer n, the inequalityMathworldPlanetmath |n2-m3|>Cm, (with m also being a positive integer and C being a number less than 1 that nears 1 as n tends to infinityMathworldPlanetmath) always holds.

The reason for the exception of perfect sixth powers (those cases of n for which there is a solution to n2=h6 in integers) is a simple consequence of associativity: if n2=h6, then h6=h2h2h2=h3h3. Then m=h and n2-m3=0. For example, 82-43=0.

For small n, C can’t be exactly 1. For example, 32-23=1, and 2>1. But even among the smaller numbers, the conjecture generally holds even with C=1. After n=3, the next counterexampleMathworldPlanetmath (that is not a perfect sixth power) to C=1 is n=378661, with the corresponding m=5234 producing a differencePlanetmathPlanetmath of just 17. A078933 in Sloane’s OEIS lists smaller values of m with cubes being at a distance from the nearest square that is less than m. Noam Elkies has found some fairly large counterexamples to setting C=1, such as n=447884928428402042307918 and m=5853886516781223, the difference between the square of the former and the cube of the latter being a relatively small 1641843.


  • 1 R. K. Guy, Unsolved Problems in Number TheoryMathworldPlanetmathPlanetmath New York: Springer-Verlag 2004: D9
Title Marshall Hall’s conjecture
Canonical name MarshallHallsConjecture
Date of creation 2013-03-22 18:15:36
Last modified on 2013-03-22 18:15:36
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 4
Author PrimeFan (13766)
Entry type Conjecture
Classification msc 11D79
Synonym Marshall Hall conjecture
Related topic PerfectPower