• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
  Representations by 2sinh^[0.5] tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 11/15/2014, 01:21 PM As mentioned before Im considering number theory connections to tetration. One of those ideas is representations. Every integer M is the sum of 3 triangular number ... or 4 squares. Also every integer M is the sum of at most O(ln(M)) powers of 2. This is all classical and pretty well known. But there are functions that have growth between polynomials and powers of 2. So since 2sinh is close to exp and 2sinh^[0.5](0) = 0 , it is natural to ask 2S(n) := floor 2sinh^[0.5](n) 2S-1(n) := floor 2sinh^[-0.5](n) 2S numbers := numbers of type 2S(n) 2S-1 numbers := numbers of type 2S-1(n) 1) Every positive integer M is the sum of at most A(M) 2S numbers. A(M) = ?? 2) Every positive integer M is the sum of at most 2S(M) B numbers. B numbers := B(n) = ?? Of course we want sharp bounds on A(M) and B(n). ( A(M) = 4 + 2^M works fine but is not intresting for instance ) regards tommy1729 tommy1729 Ultimate Fellow     Posts: 1,358 Threads: 330 Joined: Feb 2009 11/16/2014, 07:40 PM To estimate A(M) I use the following Tommy's density estimate *** Let f(n) be a strictly increasing integer function such that f(n)-f(n-1) is also a strictly increasing integer function. Then to represent a positive density of primes between 2 and M we need to take T_f(M) elements of f(n). T_f(M) is about ln(M)/ln(f^[-1](M)). This is an upper estimate. *** In this case to represent a positive density of primes between 2 and M we then need about ln(M)/ln^[3/2](M) 2S numbers. This is a brute upper estimate. A(M) is estimated as sqrt( ln(M)/ln^[3/2](M) ). Improvement should be possible. regards tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 2,372 01/17/2017, 07:21 AM Last Post: sheldonison 2sinh^[r](z) = 0 ?? tommy1729 0 1,276 02/23/2016, 11:13 PM Last Post: tommy1729  Beyond Gamma and Barnes-G tommy1729 1 2,233 12/28/2014, 05:48 PM Last Post: MphLee  Uniqueness of periodic superfunction tommy1729 0 1,861 11/09/2014, 10:20 PM Last Post: tommy1729  The angle fractal. tommy1729 1 2,161 10/19/2014, 03:15 PM Last Post: tommy1729  " statistical dynamics " tommy1729 0 1,645 08/31/2014, 11:53 PM Last Post: tommy1729  exp^[s](z) = z tommy1729 0 1,512 08/26/2014, 07:45 PM Last Post: tommy1729  composition of 3 functions. tommy1729 0 1,713 08/25/2014, 12:08 AM Last Post: tommy1729  combining some recent ideas. tommy1729 0 1,609 08/19/2014, 12:25 PM Last Post: tommy1729  Inconsistant equation ? tommy1729 0 1,728 07/27/2014, 02:38 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s) 