Matthews Map

The Matthews map generalizes the piecewise integer iteration in the Collatz conjecture. Let , let , ..., be nonzero integers, and choose integers satisfying . It is the map

(1)

when . The congruence condition ensures that is an integer.

Equivalently, the map can be written

(2)

when , where , ..., are integers and is the floor function. The map is connected with ergodic theory and Markov chains.

The Terras map, which performs the odd Collatz step and immediately divides the necessarily even result by 2, is obtained by taking , , , , and . Matthews and Watts (1984) studied how the product of the multipliers affects whether typical trajectories cycle or diverge.

Matthews obtained the following data for the mapping

(3)

where .

	number of cycles	maximum cycle length
0	5	11
1	10	27
2	13	34
3	17	118
4	19	118
5	21	165
6	23	433

Matthews and Watts (1984) proposed the following conjectures.

1. If , then all trajectories for eventually cycle.

2. If , then almost all trajectories for are divergent, except for an exceptional set of integers satisfying

(4)

3. The number of cycles is finite.

4. If the trajectory for is not eventually cyclic, then the iterates are uniformly distributed modulo for each , with

(5)

for .

Matthews conjectured that the map

(6)

either reaches a multiple of 3 or enters one of the cycles or , and offered a $100 prize for a proof (Matthews).