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From: AI on 26 Nov 2009 13:36
Prove/Disprove that nCr for (n-r) > r > 3 can never be a perfect
square.

{Note:
For r=2 we have http://www.research.att.com/~njas/sequences/A055997
which will always give perfect square nCr.
For r=3 we have n={3,4,50} to give perfect square (these are the only
three tetrahedral numbers which are also perfect square)
But for r>3 I can not find any!
}
From: Ask me about System Design on 26 Nov 2009 14:17
On Nov 26, 10:36 am, AI <vcpan...(a)gmail.com> wrote:
> Prove/Disprove that nCr for (n-r) > r > 3 can never be a perfect
> square.
>
> {Note:
> For r=2 we havehttp://www.research.att.com/~njas/sequences/A055997
> which will always give perfect square nCr.
> For r=3 we have n={3,4,50} to give perfect square (these are the only
> three tetrahedral numbers which are also perfect square)
> But for r>3 I can not find any!
>
>
>
> }- Hide quoted text -
>
> - Show quoted text -

Thought 1: Work by Finsler (I think?) led to estimates
on pi(2n) - pi(n), the number of primes between n and
2n. This may be useful in showing the results for r
not so close to n/2.

Thought 2: Use one of a number of methods to count the
power of 2 that exactly divides n C r, and then do the
same with 3. This should give conditions on n and r
that can be further refined with larger primes.

Avoiding primes between n and n-r will itself be a
challenge. Tables of prime gaps compiled so far
suggest that r will be much less that sqrt(n).
Good luck.

Gerhard "Ask Me About System Design" Paseman, 2009.11.27
From: Gerry Myerson on 26 Nov 2009 18:46
In article
<e3f65ff4-8800-486d-ba4f-fccd9640b50b(a)d9g2000prh.googlegroups.com>,
AI <vcpandya(a)gmail.com> wrote:

> Prove/Disprove that nCr for (n-r) > r > 3 can never be a perfect
> square.
>
> {Note:
> For r=2 we have http://www.research.att.com/~njas/sequences/A055997
> which will always give perfect square nCr.
> For r=3 we have n={3,4,50} to give perfect square (these are the only
> three tetrahedral numbers which are also perfect square)
> But for r>3 I can not find any!
> }

Kalman Gyory, Power values of products of consecutive integers and
binomial coefficients, in Number Theory and its Applications (Kyoto,
1997), 145-156, Kluwer, 1999, is reviewed in Math Reviews
2001a:11050. The review by Natarajan Saradha says,

A related equation treated by Erdos is
(n + k - 1) choose k = x^L in integers k > 1, n > k + 1, x, L > 1.
In 1951 Erdos showed that for k > 3 the equation has no solution.

The Erdos paper appears to be On a Diophantine equation, J London
Math Soc 26 (1951) 176-178, MR 12, 804d.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: AI on 27 Nov 2009 00:11
On Nov 27, 4:46 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <e3f65ff4-8800-486d-ba4f-fccd9640b...(a)d9g2000prh.googlegroups.com>,
>
>  AI <vcpan...(a)gmail.com> wrote:
> > Prove/Disprove that nCr for (n-r) > r > 3 can never be a perfect
> > square.
>
> > {Note:
> > For r=2 we havehttp://www.research.att.com/~njas/sequences/A055997
> > which will always give perfect square nCr.
> > For r=3 we have n={3,4,50} to give perfect square (these are the only
> > three tetrahedral numbers which are also perfect square)
> > But for r>3 I can not find any!
> > }
>
> Kalman Gyory, Power values of products of consecutive integers and
> binomial coefficients, in Number Theory and its Applications (Kyoto,
> 1997), 145-156, Kluwer, 1999, is reviewed in Math Reviews
> 2001a:11050. The review by Natarajan Saradha says,
>
> A related equation treated by Erdos is
> (n + k - 1) choose k = x^L in integers k > 1, n > k + 1, x, L > 1.
> In 1951 Erdos showed that for k > 3 the equation has no solution.
>
> The Erdos paper appears to be On a Diophantine equation, J London
> Math Soc 26 (1951) 176-178, MR 12, 804d.
>
> --
> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)

Thanks, on googling your references this is what I got
http://www1.spms.ntu.edu.sg/~guojian/MAS790/fredbeam.pdf
From: Gerry on 27 Nov 2009 06:57
On Nov 27, 4:11 pm, AI <vcpan...(a)gmail.com> wrote:
> On Nov 27, 4:46 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
> wrote:
>
> > In article
> > <e3f65ff4-8800-486d-ba4f-fccd9640b...(a)d9g2000prh.googlegroups.com>,
>
> >  AI <vcpan...(a)gmail.com> wrote:
> > > Prove/Disprove that nCr for (n-r) > r > 3 can never be a perfect
> > > square.
>
> > Kalman Gyory, Power values of products of consecutive integers and
> > binomial coefficients, in Number Theory and its Applications (Kyoto,
> > 1997), 145-156, Kluwer, 1999, is reviewed in Math Reviews
> > 2001a:11050. The review by Natarajan Saradha says,
>
> > A related equation treated by Erdos is
> > (n + k - 1) choose k = x^L in integers k > 1, n > k + 1, x, L > 1.
> > In 1951 Erdos showed that for k > 3 the equation has no solution.
>
> > The Erdos paper appears to be On a Diophantine equation, J London
> > Math Soc 26 (1951) 176-178, MR 12, 804d.
>
> Thanks, on googling your references this is what I got
> http://www1.spms.ntu.edu.sg/~guojian/MAS790/fredbeam.pdf

Great! So it appears that the result you want was proved by Erdos
in 1939, but title and journal are not given. Somewhere on the
web there's a complete list of Erdos' papers, so you could
probably track it down, if you want to.
--
GM
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