Abstract:
The principal divisors of a positive integer are its maximal prime powers, so 8 and 9 are the principal divisors of 72. Every positive integer is the product of its principal divisors. If the positive integer n has r principal divisors, we say that its principal divisor rank is r. Clearly there are no long runs of consecutive integers with principal divisor rank 1 (though there is such a run of length 4). Runs of consecutive integers with principal divisor rank 2 are more interesting (33, 34, 35, 36 is an example). And what about pairs of consecutive integers with principal divisor rank r? Are there infinitely many such pairs? More fundamentally, are there any such pairs?
I will present some recent results obtained in this area in continuing joint work with Jim MacDougall and Jason Kimberley at University of Newcastle, Australia. Be prepared to meet some rather large numbers!