A conjecture about number theory and tetration
A conjecture about number theory and tetration.

A natural number n may be factored as n = p1^a1 p2^a2 p3^a3 ... where the p_m are distinct prime numbers and a_m are natural numbers. Since the a_m are natural numbers, they may be factored in such a manner as well.

This process may be continued, building a “factorization tree” until all
the top numbers are 1.
This problem is about the height of n which we denote as h(n).
We Define:

D_n = lim N -> +oo N^-1 * |{k =< N : h(k) => n}|

D_n is sort of the density of numbers with height at least n. It is obvious that D_1 = 1 since all numbers have height at least 1.

Let a be the average height of a natural number (i.e. if you were to
pick many numbers at random their height would average out to a).
Using the previous part and other methods, give bounds on a. The
best bounds afaik [Andrew Snowden has found them] are :

1.42333 < a < 1.4618


for all n :
1/2 < D_n 2^^n =< 3

I felt the need to show the conjecture to give it more attention as it deserves.

Also some recent members might not have seen it yet and that would be a pity if they would never see it.

I also would like to see more work concerning this , for instance how Andrew found 1.42333.

If I am very optimistic it might lead to a new method for tetration.

I said very optimistic ...



Possibly Related Threads…
Thread Author Replies Views Last Post
  What is this number (\(\dots 98869612812995910644531\))? marcokrt 1 293 05/30/2023, 07:09 PM
Last Post: marcokrt
  [UFO] "commutative number theory " f(g) = g(f) tommy1729 0 219 03/24/2023, 11:36 PM
Last Post: tommy1729
  geometric function theory ideas tommy1729 0 359 12/31/2022, 12:19 AM
Last Post: tommy1729
  tommy's group addition isomo conjecture tommy1729 1 617 09/16/2022, 12:25 PM
Last Post: tommy1729
  [NT] primitive root conjecture tommy1729 0 483 09/02/2022, 12:32 PM
Last Post: tommy1729
Question Is the Tetra-Euler Number Rational? Catullus 1 660 07/17/2022, 06:37 AM
Last Post: JmsNxn
Question Natural Properties of the Tetra-Euler Number Catullus 6 1,702 07/01/2022, 08:16 AM
Last Post: Catullus
  tommy's new conjecture/theorem/idea (2022) ?? tommy1729 0 580 06/22/2022, 11:49 PM
Last Post: tommy1729
  " x-theory " tommy1729 1 1,442 08/12/2021, 12:17 AM
Last Post: tommy1729
  Dynamical Systems and Number Theory Daniel 4 3,243 06/01/2021, 11:34 PM
Last Post: JmsNxn

Users browsing this thread: 1 Guest(s)