Problem of the Week 1054

Wild and Wooley Numbers

Can 5 be written as a product of fractions in this set?

W = {2/1, 5/3, 8/5, 11/7, 14/9 ...} = {(3n + 2)/(2n + 1) : n ≥ 0}

Repetitions are allowed.

Note: Extra credit: Same question for 10 and 20.

Source. J. Lagarias, Amer. Math. Monthly. Feb 2006.

Lagarias's paper talks about wild numbers and Wooley numbers (latter named for Trevor D. Wooley). The set of numbers expressible in terms of W given above is the Wooley set. If the fraction 1/2 is added to the generators, then the expressible numbers are called "wild." Questions about these sets are related to the notorious 3x + 1 problem. Recently Applegate and Lagarias have proved the following noteworthy theorem: All integers except those divisible by 3 are in the wild set.

© Copyright 2006 Stan Wagon. Reproduced with permission.

21 March 2006