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:

$[2^3,3^2],[5^2,3^3],[2^5,6^2],[11^2,5^3],[3^7,13^3],$

$[5^5,56^2],[181^2,2^{15}],[43^3,282^2],[46^3,312^2],[22434^2,55^5].$

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)