R.P. Steiner. "A theorem on the Syracuse problem". In: ed. by D. McCarthy and H. C. Williams. Congressus numerantium; 20. Proceedings of the 7th Manitoba Conference on Numerical Mathematics and Computation, September 29-October 1, 1977. Winnipeg: Utilitas Mathematica Pub., 1978, pp. 553-559.
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I think there is no open exemplar on the net (I've searched a couple of times intensely).
But the articles of John Simons/Benne de Weger (linked to by wikipedia) from 2000 to 2002 on the $2$-cycle and the $m$-cycle problem refer to it fairly explicite.
Moreover, I've a personal mail of R. Steiner where he sketched his proof-idea for me and it is pretty short: the key seems to be the idea of introducing this Baker-style argument at all. And then the sharpening of bounds by a later author (Mignotte?) which in connection with the continued-fraction-convergents allowed to complete the proof via accessible range of numbers.
Gottfried Helms
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I presume the "sketch" you talk of is only a sketch of a proof of something weaker than the conjecture itself, is it? – it's a hire car baby Mar 17 '19 at 17:01
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@user334732: It is a sketch (an informal mail) of the essentials of his proof, that there cannot exist any "1-cycle" except the trivial one. For definition what a "1-cycle" is, please read wikipedia or some of my recent answers to recent collatz-tagged-questions here. – Gottfried Helms Mar 17 '19 at 17:06
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1Thanks. I believe a 1-cycle goes up then down, withonly one change of direction, when seen using the ${(3x+1)/2,x/2)$ version of the problem. Which I once before misinterpreted for just goes up then down in the ${3x+1,x/2}$ version - which is of course trivial to prove. – it's a hire car baby Mar 17 '19 at 17:10
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Not sure if this is what you want, but a Google search for "R.P. Steiner. "A theorem on the Syracuse problem"." returned this as the first link:
https://www.sciencedirect.com/science/article/pii/S0022314X06001223
The paper is freely downloadable.
marty cohen
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1Right. This paper only references the Steiner paper. I actually already have this paper. – MathAllTheTime Jan 29 '19 at 21:35
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1@MathAllTheTime: were you able to send me a copy of the Steiner-article? – Gottfried Helms Mar 17 '19 at 17:08
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1@GottfriedHelms I just e-mailed it to you. Let me know if you do not receive. It is 7 separate emails, one for each page. – MathAllTheTime Mar 18 '19 at 19:20
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1I am a graduate student at University of Bonn and I also would like to read the paper, can somebody mail it to me? – sdigr Jul 17 '20 at 11:09
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@GottfriedHelms Is there any chance you still have the Steiner paper and could send it to me? I need it for a school project. I would ask MathAllTheTime, but it seems he hasn't been online in over two years. Thanks. – William Ryman Jan 09 '22 at 16:38