Marshall Hall’s conjecture
Conjecture. (Marshall Hall, Jr.). With the exception of ${n}^{2}$ being a perfect sixth power, for any positive integer $n$, the inequality^{} $|{n}^{2}-{m}^{3}|>C\sqrt{m}$, (with $m$ also being a positive integer and $C$ being a number less than 1 that nears 1 as $n$ tends to infinity^{}) always holds.
The reason for the exception of perfect sixth powers (those cases of $n$ for which there is a solution to ${n}^{2}={h}^{6}$ in integers) is a simple consequence of associativity: if ${n}^{2}={h}^{6}$, then ${h}^{6}={h}^{2}{h}^{2}{h}^{2}={h}^{3}{h}^{3}$. Then $m=h$ and ${n}^{2}-{m}^{3}=0$. For example, ${8}^{2}-{4}^{3}=0$.
For small $n$, $C$ can’t be exactly 1. For example, ${3}^{2}-{2}^{3}=1$, and $\sqrt{2}>1$. But even among the smaller numbers, the conjecture generally holds even with $C=1$. After $n=3$, the next counterexample^{} (that is not a perfect sixth power) to $C=1$ is $n=378661$, with the corresponding $m=5234$ producing a difference^{} 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 $\sqrt{m}$. 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.
References
- 1 R. K. Guy, Unsolved Problems in Number Theory^{} 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 |