Redmond-Sun conjecture

Conjecture. (Stephen Redmond & Zhi-Wei 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	Redmond-Sun conjecture
Canonical name	RedmondSunConjecture
Date of creation	2013-03-22 17:26:50
Last modified on	2013-03-22 17:26:50
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	7
Author	PrimeFan (13766)
Entry type	Conjecture
Classification	msc 11N05