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

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

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Erick Wong

Posts: 394
Registered: 12/8/04
Re: consecutive square-free integers
Posted: Nov 23, 2001 5:13 PM
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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

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