The Collatz Problem in the Light of an Infinite Free Semigroup

The Collatz (or <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>) problem is examined in terms of a free semigroup on which suitable diophantine and rational functions are defined. The elements of the semigroup, called T-words, comprise the information about the Collatz operations which relate an odd start number to an odd end number, the group operation being the concatenation of T-words. This view puts the concept of encoding vectors , first introduced in 1976 by Terras , in the proper mathematical context. A method is described which allows to determine a one-parameter family of start numbers compatible with any given T-word. The result brings to light an intimate relationship between the Collatz <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math> problem and the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math> problem. Also, criteria for the rise or fall of a Collatz sequence are derived and the important notion of anomalous T-words is established. Furthermore, the concept of T-words is used to elucidate the question what kind of cycles—trivial, nontrivial, rational—can be found in the Collatz <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math> problem and also in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>3</mml:mn><mml:mi>m</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:math> problem. Furthermore, the notion of the length of a Collatz sequence is discussed and applied to average sequences . Finally, a number of conjectures are proposed.