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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:
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Conjecture. (Stephen Raymond \& 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:
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| \begin{enumerate} |
\begin{enumerate} |
| \item There are no primes between $2^3$ and $3^2$. |
\item There are no primes between $2^3$ and $3^2$. |
| \item There are no primes between $5^2$ and $3^3$. |
\item There are no primes between $5^2$ and $3^3$. |
| \item There are no primes between $2^5$ and $6^2$. |
\item There are no primes between $2^5$ and $6^2$. |
| \item There are no primes between $11^2$ and $5^3$. |
\item There are no primes between $11^2$ and $5^3$. |
| \item There are no primes between $3^7$ and $13^3$. |
\item There are no primes between $3^7$ and $13^3$. |
| \item There are no primes between $5^5$ and $56^2$. |
\item There are no primes between $5^5$ and $56^2$. |
| \item There are no primes between $181^2$ and $2^{15}$. |
\item There are no primes between $181^2$ and $2^{15}$. |
| \item There are no primes between $43^3$ and $282^2$. |
\item There are no primes between $43^3$ and $282^2$. |
| \item There are no primes between $46^3$ and $312^2$. |
\item There are no primes between $46^3$ and $312^2$. |
| \item There are no primes between $22434^2$ and $55^5$. |
\item There are no primes between $22434^2$ and $55^5$. |
| \end{enumerate} |
\end{enumerate} |
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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 $55^5$.
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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 counteexamples have been found past $55^5$.
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