
Re: consecutive squarefree integers
Posted:
Nov 23, 2001 5:13 PM


Erick Bryce Wong <erick@sfu.ca> wrote: >Bill Taylor <mathwft@math.canterbury.ac.nz> wrote: >>kramsay@aol.commangled (Keith Ramsay) writes: >>Are there infinitely many cases of n, n+1, n+2 all being squarefree? >> >>Would be interesting to know. I guess "yes", on probabilistic grounds... > >Yes, by a simple refinement of the same density argument. Consider Z\4Z, [snip] >I'm surprised this argument isn't more wellknown; I haven't seen it written >anywhere, hence my somewhat poor presentation :).
Okay, to put it much more elegantly...Suppose that for sufficiently large n, {4n+1, 4n+2, 4n+3} contained at most 2 squarefree integers. Since 4n is never squarefree, this would imply the upper density of the squarefree integers is at most 1/2, which is not true.
 Erick

