Lagarias 2003 annotated bibliography 1

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conjecture_p1: The 3x+1 Conjecture as the bibliography's abstract and introduction pose it, for the 3x+1 function T, which is the map of Problem 1135, with the Collatz function C defined beside it and the conjecture recorded as unsolved.

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Jeffrey C. Lagarias, The 3x+1 Problem: An Annotated Bibliography (1963-1999) (Sorted by Author), arXiv:math/0309224v13 (2011 revision; 74 pp.).

One annotated entry per work of the period, sorted by author; the bibliography proves nothing of its own. Its abstract and the opening of §1 (p. 1; §1 runs to p. 2) define the $3x+1$ function $T$ and the Collatz function $C$ and pose the $3x+1$ Conjecture, recorded on the conjecture page. The erdosproblems.com page for problem 1135 cites other works of Lagarias on the problem, not this bibliography.

The copy read for this card is arXiv:math/0309224v13 (dated January 1, 2011; 74 pp.). The arXiv record names arXiv's non-exclusive distribution license (arXiv:math/0309224), every other right reserved.

Read status: claims checked, for the statements the bibliography makes; it proves nothing. The abstract and §1 (pp. 1--2), entries 55 and 56 (pp. 23--24) and entry 77 (pp. 30--31) were read on the page images.

Bears on.

• #1135: the problem's map $f$ is the bibliography's $T$ on the positive integers, and its question is the bibliography's $3x+1$ Conjecture (p. 1), which the abstract records as unsolved; see the conjecture page.
• #1134: entries 55 and 56 (pp. 23--24), the annotations of the Erdős--Graham 1979 paper and of their 1980 monograph, each report that Erdős showed (see Klarner and Rado (1974)) that $\sum1/a_i<1$ forces density $0$ for the set generated by maps $a_1x+b_1,\dots,a_rx+b_r$, that the case $\sum1/a_i=1$ "is pointed to as a source of unresolved problems", and, quoted: "Erdős had proposed as a prize problem: For $R=\{2x+1,3x+1,6x+1\}$ and $A=\{1\}$ is the set $<R:A>$ of positive density? This was answered in the negative by D. J. Crampin and A. J. W. Hilton (unpublished), as summarized in Klarner (1982) and Klarner (1988)." That prize problem is the problem's question with the same three maps, asked for positive density where the problem asks for positive lower density; the bibliography reports the negative answer as unpublished and names only Klarner's summaries. Entry 77 (pp. 30--31) is Guy (1983a), the work the problem page lists as [Gu83b]; its annotation says that Guy's Problem 4 asks a question it attributes to Klarner (1982), with the maps $x\mapsto2x$, $x\mapsto3x+2$, $x\mapsto6x+3$ and positive lower density asked, which is not the problem's set. The problem page's references do not include this bibliography.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.