32
Gerhard Opfer
Table 5.5. Exponents of the general power expansion of h satisfying V [h] = 0 ordered backwards, starting with powers of two. Entries m with m + 1 = 3k in red. 4 8 5 3 16 32 21 64 128 85 256 512 341 227 151 1024 2048 1365 4096 8192 5461 16384 32768 21845 14563 65536 131072 87381 262144 524288 349525 1048576 Acknowledgment. This research was supported by the German Science Foundation, DFG, GZ: OP 33/19-1. The author would like to thank Lothar Berg (Rostock) and Bodo Werner (Hamburg) for many valuable comments. The paper was completed on Thursday, May 5, 2011, which is Cinco de Mayo, a Mexican holiday. REFERENCES [1] L. Berg and G. Meinardus, Functional equations connected with the Collatz problem, Results Math., 25 (1994), pp. 1–12. [2] L. Berg and G. Meinardus, The 3n + 1 Collatz problem and functional equations, Rostock. Math. Kolloq. 48 (1995), pp. 11–18. ¨ [3] L. Collatz, Uber die Entstehung des (3n + 1) Problems, J. Qufu Norm. Univ. Nat. Sci. Ed., 3 (1986), pp. 9–11 (in Chinese).1 [4] R. K Guy, Don’t try to solve these problems, Amer. Math. Monthly, 90 (1983), pp. 35–41. [5] J. C. Lagarias, ed., The Ultimate Challenge. The 3x + 1 Problem, Amer. Math. Soc., 2010, 344 pp.2 ¨ [6] G. Meinardus, Uber das Syracuse-Problem, Preprint Nr. 67, Universit¨ at Mannheim, 1987, 32 p. ¨ [7] G. J. Wirsching, Uber das 3n + 1 Problem, Elem. Math. 55 (2000), pp. 142–155.
1 The
German title as used here was handwritten by Collatz on a copy of the Chinese paper. the time of the submission (May 25, 2011), this book, announced by the publisher, was not yet available to the author. 2 At