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Redmond-Sun conjecture (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 \neq 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$.



"Redmond-Sun conjecture" is owned by PrimeFan.
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Cross-references: counterexamples, ranges, perfect, OEIS, primes, numbers, exponents, integers, positive, Zhi-Wei Sun, conjecture

This is version 4 of Redmond-Sun conjecture, born on 2007-08-02, modified 2007-08-19.
Object id is 9828, canonical name is RedmondSunConjecture.
Accessed 2072 times total.

Classification:
AMS MSC: 11N05 (Number theory :: Multiplicative number theory :: Distribution of primes)
Pending Errata and Addenda
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