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}],}](../I/m/f95d06cd40ce12e38f805546202bb1309d20ce6a.svg)
![{\displaystyle [5^{5},\,56^{2}],\ [181^{2},\,2^{15}],\ [43^{3},\,282^{2}],\ [46^{3},\,312^{2}],\ [22434^{2},\,55^{5}].}](../I/m/bd30c1e6ff3e8bbd105434ee5e270f4003b44a52.svg)
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.
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