[2014] exp^[s](z) = z - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: [2014] exp^[s](z) = z (/showthread.php?tid=916) [2014] exp^[s](z) = z - tommy1729 - 08/26/2014 Consider the equation for a given real s > 0 : exp^[s](z) = z such that |z| < exp^[s](s). Let the number of distinct solutions be T[s]. Because of the fact that if w satisfies exp^[s](w) = w then conj(w) = w_ satisfies exp^[s](w_) =w_ Hence we can conclude that T[s] is Always even. Therefore I am intrested in T[s]/2. I would love to see plots and tables of s vs T[s]/2. Many conjectures are possible. Probably connected to fractal theory , basic dynamical systems and bifurcations. Being very optimistic I would say T[s]/2 might have a closed form. Possible conjectures could look like : 1) T[s]/2 = O ( s^a b^s ) for some real a,b. 2) Let p be an odd prime such that p+2 is not a prime. Then T[p] =< T[p+2]. Perhaps someone here is an expert on these things ? Let n be a positive integer. Could T[n] satisfy a recursion ? Like T[2n] = T[2n-1] + T[n] :p regards tommy1729