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'Redmond-Sun conjecture'
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| Title of object: |
Redmond-Sun conjecture |
| Canonical Name: |
RedmondSunConjecture |
| Type: |
Conjecture |
| Created on: |
2007-08-02 18:57:52 |
| Modified on: |
2007-08-02 18:57:52 |
| Classification: |
msc:11P32 |
Revision comment (for changes between this and next version):
| Changed MSC from 11P32 to 11N05 per correction #12947 ('Wrong MSC'). |
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Content:
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:
\begin{enumerate}
\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 $2^5$ and $6^2$.
\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 $5^5$ and $56^2$.
\item There are no primes between $181^2$ and $2^15$.
\item There are no primes between $43^2$ and $282^2$.
\item There are no primes between $46^3$ and $312^2$.
\item There are no primes between $22434^2$ and $55^5$.
\end{enumerate}
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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