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

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