Marshall Hall’s conjecture
Conjecture. (Marshall Hall, Jr.). With the exception of being a perfect sixth power, for any positive integer , the inequality , (with also being a positive integer and being a number less than 1 that nears 1 as tends to infinity) always holds.
For small , can’t be exactly 1. For example, , and . But even among the smaller numbers, the conjecture generally holds even with . After , the next counterexample (that is not a perfect sixth power) to is , with the corresponding producing a difference of just 17. A078933 in Sloane’s OEIS lists smaller values of with cubes being at a distance from the nearest square that is less than . Noam Elkies has found some fairly large counterexamples to setting , such as and , 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 Theory New York: Springer-Verlag 2004: D9
|Title||Marshall Hall’s conjecture|
|Date of creation||2013-03-22 18:15:36|
|Last modified on||2013-03-22 18:15:36|
|Last modified by||PrimeFan (13766)|
|Synonym||Marshall Hall conjecture|