Redmond–Sun conjecture

In mathematics, the Redmond–Sun conjecture, raised by Stephen Redmond and Zhi-Wei Sun in 2006, states that every interval [x ^m, y ⁿ] with x, y, m, n ∈ {2, 3, 4, ...} and x ^m ≠ y ⁿ contains primes with only finitely many exceptions. Namely, those exceptional intervals [x ^m, y ⁿ] are as follows:

[math]\displaystyle{ [2^3,\,3^2],\ [5^2,\,3^3],\ [2^5,\,6^2],\ [11^2,\,5^3],\ [3^7,\,13^3], }[/math]

[math]\displaystyle{ [5^5,\,56^2],\ [181^2,\,2^{15}],\ [43^3,\,282^2],\ [46^3,\,312^2],\ [22434^2,\,55^5]. }[/math]

The conjecture has been verified for intervals [x ^m, y ⁿ] with endpoints below 4.5 x 10¹⁸. It includes Catalan's conjecture and Legendre's conjecture as special cases. Also, it is related to the abc conjecture as suggested by Carl Pomerance.

External links

• Redmond-Sun conjecture at PlanetMath.org.
• Number Theory List (NMBRTHRY Archives) --March 2006
• OEIS sequence A116086 (Perfect powers n with no primes between n and the next larger perfect power)