The 3x+ 1 problem concerns iteration of the map on the integers given by T(n) = (3n+1)/2 if n is odd; T(n) = n/2 if n is even. The 3x+1 Conjecture asserts that for every positive integer n > 1 the forward orbit of n under iteration by T includes the integer 1. This paper is an annotated bibliography of work done on the 3x+1 problem and related problems from 1963 through 1999. At present the 3x+1 Conjecture remains unsolved.

Two Conjectures or how to win £1100

The 3x+1 problem: An annotated bibliography

We describe an improved formalisation of slicing in WSL transformation theory and apply the result to a particularly challenging slicing problem: the SCAM mug (Anon, 2001). We present both syntactic and semantic slices of the mug program and give semantic slices for various generalisations of the program. Although there is no algorithm for constructing a minimal syntactic slice, we show that it is possible, in the WSL language, to derive a minimal semantic slice for any program and any slicing criteria.

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Slicing the SCAM mug: a case study in semantic slicing

Theorem 1.1 Every integer string of length n will iterate to the zero-string in a finite number steps if and only if n 1⁄4 2 for some positive integer m. Remarkably, this theorem has been restated and reproved (using various techniques) a number of times since then; see Refs. [3,8,18,19,22,23,25,27]. Other references may be found in Ref. [17]. This process has been most commonly referred to as a Ducci sequence or the N -number game. Other interesting questions and observations concerning the iteration of f have arisen. Lotan [11] showed that every 4-string with real entries iterates to the zero-string in finitely many steps except for strings (up to shifts, reflection and scaling) of the form ð1; q; q; q3Þ where q 8 1:839 is a root of q 2 q 2 q 2 1 1⁄4 0. These exceptional strings iterate to the zero-string asymptotically. Work towards this result was also pursued geometrically in Ref. [16]. Webb [24] showed that 4-strings may take arbitrarily long to reach the zero-string; if one considers the Tribonacci numbers defined by tn 1⁄4 tn21 þ tn22 þ tn23; t0 1⁄4 0; t1 1⁄4 1; t2 1⁄4 1;

Unbounded Ducci Sequences

Ducci n-number games have been studied over the last century by a variety of researchers. This article considers the Ducci Game as a map over the finite field ZZ2. The period lengths are characterized through orders of minimal annihilating polynomials. The minimal polynomial of the map can be explicitly written down for any n. Using the minimal polynomial, we provide algebraic proofs of divisibility conditions, initially obtained in Ehrlich. In addition, data on all periodic cycles for vector lengths up to n = 40 are computed.

/pdf/a-characterization-for-the-length-of-cycles-of-the-n-number-2mow8h5i51.pdf

A characterization for the length of cycles of the n - number ducci game

In 1996, Sir Bryan Thwaites[4] posed two open problems with prize money offered for solutions. The first problem (with a £1000 reward) is the wellknown 3x + 1 problem which has received attention from many quarters. This easily-stated problem has alluded mathematicians for about fifty years; for more information, see Lagarias[1] and Wirsching[5]. Thwaites’s other problem (with a £100 reward) has no clear origin. He states it as follows:

background

Iterated Strings and Cellular Automata.

/pdf/two-conjectures-or-how-to-win-1100-3r9hdm0apd.pdf

Two Conjectures or how to win £1100

Citations

The 3x+1 problem: An annotated bibliography

Slicing the SCAM mug: a case study in semantic slicing

Unbounded Ducci Sequences

A characterization for the length of cycles of the n - number ducci game

Iterated Strings and Cellular Automata.

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