## Primes between squares

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• Is there always a prime between x^2 and (x+n)^2? Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry Brainbench Most Valuable
Message 1 of 8 , May 18, 2001
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Is there always a prime between x^2 and (x+n)^2?

Jon Perry
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• Is there always a prime between x^2 and (x+n)^2? Jon Perry A casual glance would suggest that there is. Furthermore, it would seem that the number of primes
Message 2 of 8 , May 18, 2001
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Is there always a prime between x^2 and (x+n)^2?

Jon Perry

A casual glance would suggest that there is. Furthermore, it would seem that the number of primes between x^2 and (x+1)^2 increases as x increases. I ran through the cases from x=1 to x=19. The number of primes between starts out with an average of 2 and ends with an average of 6 when x=19.
Ron McCluskey

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• ... The answer is almost certainly yes, but it s unproven. See the last entry on the page: http://www.utm.edu/research/primes/notes/conjectures/
Message 3 of 8 , May 18, 2001
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--- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
> Is there always a prime between x^2 and (x+n)^2?
>

The answer is almost certainly yes, but it's unproven.

See the last entry on the page:

http://www.utm.edu/research/primes/notes/conjectures/
• Thanks. I just wondered if the easier case had been solved for any n, including functions, etc... For example, is there always a prime between x^2 and (x+x)^2,
Message 4 of 8 , May 20, 2001
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Thanks. I just wondered if the easier case had been solved for any n,
including functions, etc...

For example, is there always a prime between x^2 and (x+x)^2, or even
(x^x)^2.

Someone stated that this was not true for n=0, which is true. As it is true
for n=1 (supposedly), this would imply that there exists a real y such that
y is minimum and there is always a prime between x^2 and (x+y)^2.

Jon Perry
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-----Original Message-----
From: jack@... [mailto:jack@...]
Sent: 18 May 2001 20:43
Subject: [PrimeNumbers] Re: Primes between squares

--- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
> Is there always a prime between x^2 and (x+n)^2?
>

The answer is almost certainly yes, but it's unproven.

See the last entry on the page:

http://www.utm.edu/research/primes/notes/conjectures/

Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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• I was trying to find out about the theorem that there is at least one prime between every pair of consecutive squares. I thought I saw someting about a proof
Message 5 of 8 , Apr 26, 2005
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I was trying to find out about the theorem that there is at least one
prime between every pair of consecutive squares. I thought I saw
someting about a proof in Archives of Mathematics in 2000, but I
theorem?

Thom
• ... I know it s not the best source, but according to `Proofs from the Book , whose last edition dates to 2000, the problem is still open. Décio [Non-text
Message 6 of 8 , Apr 27, 2005
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On Tuesday 26 April 2005 20:33, you wrote:
> I was trying to find out about the theorem that there is at least one
> prime between every pair of consecutive squares. I thought I saw
> someting about a proof in Archives of Mathematics in 2000, but I
> theorem?
>
> Thom

I know it's not the best source, but according to `Proofs from the Book',
whose last edition dates to 2000, the problem is still open.

Décio

[Non-text portions of this message have been removed]
• ... least one ... about this ... Book , ... OK, so it hasn t been proven yet. What about computer testing? I recently wrote a program to test this theorem,
Message 7 of 8 , Apr 28, 2005
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--- In primenumbers@yahoogroups.com, Décio Luiz Gazzoni Filho
<decio@d...> wrote:
> On Tuesday 26 April 2005 20:33, you wrote:
> > I was trying to find out about the theorem that there is at
least one
> > prime between every pair of consecutive squares. I thought I saw
> > someting about a proof in Archives of Mathematics in 2000, but I
> > couldn't find the paper. Does anybody have any information
> > theorem?
> >
> > Thom
>
> I know it's not the best source, but according to `Proofs from the
Book',
> whose last edition dates to 2000, the problem is still open.
>
> Décio
>
>
> [Non-text portions of this message have been removed]

OK, so it hasn't been proven yet. What about computer testing? I
recently wrote a program to test this theorem, and was wondering how
high I should start testing.

Thom
• ... You would have to start above 2*10^17 but I suggest you forget about it. All large prime gaps between smaller primes have been computed:
Message 8 of 8 , Apr 28, 2005
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thomas_ruley wrote:

> OK, so it hasn't been proven yet. What about computer testing? I
> recently wrote a program to test this theorem, and was wondering how
> high I should start testing.

You would have to start above 2*10^17 but I suggest you forget about it.
All large prime gaps between smaller primes have been computed:
http://www.ieeta.pt/~tos/gaps.html

Also see First occurrence prime gaps:
http://www.trnicely.net/gaps/gaplist.html#MainTable

The largest gap below 2*10^17 is only 1220.
Computing a counter example to primes between squares seems highly unlikely
to me.

--
Jens Kruse Andersen
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