The 3n+1 Problem & Beatty Sequences
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10-09-2014, 09:56 PM
(This post was last modified: 10-09-2014 10:08 PM by Thomas Klemm.)
Post: #6
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RE: The 3n+1 Problem & Beatty Sequences
From Wikipedia 6.3 As a parity sequence I followed the footnote [19] to find a rather old paper:
Terras, Riho (1976), "A stopping time problem on the positive integers". The theorem 1.1 (Remainder representation) provides a formula to calculate the last value in the modified Collatz sequence: \[ T^kn=\lambda_k(n)n+\rho_k(n) \] where \(\lambda_i(n)=\frac{3^{S_i(n)}}{2^i}\) and \(\rho_k=\frac{\lambda_k}{2}(\frac{X_0}{\lambda_1}+\frac{X_1}{\lambda_2}+\ldots+\frac{X_{k-1}}{\lambda_k})\). Don't let you confuse by the formula. The important thing is that \(\lambda_k(n)\) is a factor < 1 and \(\rho_k(n)\) is considered small. With a little handwaving I'd say that the factor is the result of the multiplication by 3 and division by 2 and the remainder keeps track of the occasional addition of 1. In case of the example of your slides in the video \(i=10\) and \(S_i(n)=6\) thus \(\lambda_k(n)=\frac{3^6}{2^{10}}\approx 0.7119140625\). I just calculated \(\lambda_k(n)n\) and dare to say that the approximation isn't bad: Code: 507 1101101100 362 ≈ 360.940 This factor \(\lambda_k(n)\) must be < 1 of course. Thus we have \(\frac{3^p}{2^q}<1\Rightarrow 3^p<2^q\Rightarrow p\log(3)<q\log(2)\Rightarrow p\log(3)-q\log(2)<0\Rightarrow \frac{\log(3)}{\log(2)}p-q<0\). But that's exactly the value that you calculate by the method you described. Cool that you figured that out by yourself. Don't be disappointed that it's already known. At least to me it was new and I enjoyed your video a lot. Cheers Thomas |
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Messages In This Thread |
The 3n+1 Problem & Beatty Sequences - Joe Horn - 08-26-2014, 03:13 AM
RE: The 3n+1 Problem & Beatty Sequences - Jim Horn - 08-26-2014, 09:45 PM
RE: The 3n+1 Problem & Beatty Sequences - Joe Horn - 08-28-2014, 07:03 PM
RE: The 3n+1 Problem & Beatty Sequences - Curlytop - 08-28-2014, 08:45 PM
RE: The 3n+1 Problem & Beatty Sequences - Jim Horn - 08-28-2014, 09:15 PM
RE: The 3n+1 Problem & Beatty Sequences - Thomas Klemm - 10-09-2014 09:56 PM
RE: The 3n+1 Problem & Beatty Sequences - Joe Horn - 10-10-2014, 03:11 AM
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