The Redmond – Sun hypothesis , expressed by Stefan Redmond and Zhiwei Sun in 2006, states that any interval [ x m , y n ], where x , y , m , n ∈ {2, 3, 4, ...}, contains prime numbers with a finite number of exceptions. Here are the intervals [ x m , y n ] for which the hypothesis fails:
![{\ displaystyle [2 ^ {3}, \, 3 ^ {2}], \ [5 ^ {2}, \, 3 ^ {3}], \ [2 ^ {5}, \, 6 ^ {2 }], \ [11 ^ {2}, \, 5 ^ {3}], \ [3 ^ {7}, \, 13 ^ {3}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f95d06cd40ce12e38f805546202bb1309d20ce6a)
![{\ displaystyle [5 ^ {5}, \, 56 ^ {2}], \ [181 ^ {2}, \, 2 ^ {15}], \ [43 ^ {3}, \, 282 ^ {2 }], \ [46 ^ {3}, \, 312 ^ {2}], \ [22434 ^ {2}, \, 55 ^ {5}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd30c1e6ff3e8bbd105434ee5e270f4003b44a52)
The hypothesis has been tested for intervals [ x m , y n ] below 4.5 x 10 18 . The hypothesis includes, as particular cases, the Catalan hypothesis and the Legendre hypothesis . It is also associated with the abc hypothesis , as Karl Pomerance suggests.
Links
- Redmond-Sun conjecture on PlanetMath .
- Number Theory List (NMBRTHRY Archives) --March 2006
- Sequence
A116086 in the Encyclopedia of Integer Sequences