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:

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

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

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