*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]

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