Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[2014] Representations by 2sinh^[0.5]
#1
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
Reply


Messages In This Thread
[2014] Representations by 2sinh^[0.5] - by tommy1729 - 11/15/2014, 01:21 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 2,168 01/17/2017, 07:21 AM
Last Post: sheldonison
  2sinh^[r](z) = 0 ?? tommy1729 0 1,178 02/23/2016, 11:13 PM
Last Post: tommy1729
  [2014] Beyond Gamma and Barnes-G tommy1729 1 2,078 12/28/2014, 05:48 PM
Last Post: MphLee
  [2014] Uniqueness of periodic superfunction tommy1729 0 1,730 11/09/2014, 10:20 PM
Last Post: tommy1729
  [2014] The angle fractal. tommy1729 1 1,999 10/19/2014, 03:15 PM
Last Post: tommy1729
  [2014] " statistical dynamics " tommy1729 0 1,539 08/31/2014, 11:53 PM
Last Post: tommy1729
  [2014] exp^[s](z) = z tommy1729 0 1,415 08/26/2014, 07:45 PM
Last Post: tommy1729
  [2014] composition of 3 functions. tommy1729 0 1,610 08/25/2014, 12:08 AM
Last Post: tommy1729
  [2014] combining some recent ideas. tommy1729 0 1,514 08/19/2014, 12:25 PM
Last Post: tommy1729
  [2014] Inconsistant equation ? tommy1729 0 1,609 07/27/2014, 02:38 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)