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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

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

6. 6.

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

7. 7.

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

8. 8.

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

9. 9.

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

10. 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 $55^{5}$.