Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: consecutive square-free integers
Replies: 13   Last Post: Nov 24, 2001 5:40 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Erick Wong

Posts: 394
Registered: 12/8/04
Re: consecutive square-free integers
Posted: Nov 23, 2001 5:13 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply



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 square-free?
>>
>>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 well-known; 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







Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.