Recent Publications

Almost cyclic groups A group G is almost cyclic if there is an element x in G such that for every element g in G, there is a power of x which is conjugate to g. Almost cyclic groups arose in work of W. Ziller on closed geodesics, as well as being the subject of a problem in the Kourovka notebook. The principal results of this paper are that solvable almost cyclic groups are cyclic, and one-relator almost cyclic groups are cyclic. (7 pages) [PostScript, 225214 bytes]
[May, 2009] A reviewer pointed out easier proofs of the results on solvable and one-relator groups. So I'll need to make substantial additions to this before resubmitting it. I'm leaving the old version here in the meantime.
Solution to Problem 11023 (American Mathematical Monthly) (submitted August, 2003) Wu Wei Chao asked for all integer solutions (x,y) to the equation x^2 + 3xy + 4006(x + y) + 2003^2 = 0. I consider the more general problem of finding solutions to x^2 + 3xy + 2p(x + y) + p^2 = 0, where p is prime. I show that there is one solution if p = 3, and there are two solutions otherwise (in which case explicit formulas are given for the solutions). (2 pages) [PostScript, 99152 bytes]

Last updated: May 18, 2009