Redmond-Sun conjecture

In mathematics, the Redmond-Sun conjecture (raised by Stephen Redmond and Zhi-Wei Sun in 2006) states that an interval $[x^m,y^n]$ with $x,y,m,nin\{2,3,ldots\}$ 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 $[x^m,y^n]$ below $10^\{12\}$. 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

* [http://planetmath.org/encyclopedia/RedmondSunConjecture.html Redmond-Sun Conjecture in PlanetMath]

* [http://listserv.nodak.edu/cgi-bin/wa.exe?A1=ind0603&L=nmbrthry Number Theory List (NMBRTHRY Archives) --March 2006]

* [http://www.research.att.com/~njas/sequences/A116086 Sequence A116086 in the On-Line Encyclopedia of Integer Sequences]