RedmondSun conjecture
Conjecture. (Stephen Redmond & ZhiWei Sun) Given positive integers $x$ and $y$, and exponents^{} $a$ and $b$ (with all these numbers being greater than 1), if ${x}^{a}\ne {y}^{b}$, then between ${x}^{a}$ and ${y}^{b}$ there are always primes, with only the following ten exceptions:

1.
There are no primes between ${2}^{3}$ and ${3}^{2}$.

2.
There are no primes between ${5}^{2}$ and ${3}^{3}$.

3.
There are no primes between ${2}^{5}$ and ${6}^{2}$.

4.
There are no primes between ${11}^{2}$ and ${5}^{3}$.

5.
There are no primes between ${3}^{7}$ and ${13}^{3}$.

6.
There are no primes between ${5}^{5}$ and ${56}^{2}$.

7.
There are no primes between ${181}^{2}$ and ${2}^{15}$.

8.
There are no primes between ${43}^{3}$ and ${282}^{2}$.

9.
There are no primes between ${46}^{3}$ and ${312}^{2}$.

10.
There are no primes between ${22434}^{2}$ and ${55}^{5}$.
See A116086 in Sloane’s OEIS for a listing of the perfect powers^{} beginning primeless ranges before the next perfect power. As of 2007, no further counterexamples have been found past ${55}^{5}$.
Title  RedmondSun conjecture 

Canonical name  RedmondSunConjecture 
Date of creation  20130322 17:26:50 
Last modified on  20130322 17:26:50 
Owner  PrimeFan (13766) 
Last modified by  PrimeFan (13766) 
Numerical id  7 
Author  PrimeFan (13766) 
Entry type  Conjecture 
Classification  msc 11N05 