A set A of (distinct) positive integers is "special above k" if every x in A such that x > k satisfies:

- x divides the product of all y in A with y < x
- x does not divide any y in A with y > x.
Example: {1, 2, 3, 6, 9} is special above 3 (this is the largest set that is special above 3) and {1, 2, 4, 8} is special above 4 but not special above 3.

Find as large a set as you can that is special above 4.

Find as large a set as you can that is special above 5.Feel free to investigate farther... the shocking details of what is known will come next week.

Source: Harvey Friedman, Ohio State University© Copyright 2000 Stan Wagon. Reproduced with permission.

26 September 2000