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From: Gerry Myerson on 29 Nov 2009 17:43
In article
<5a3fd17c-4f48-4245-8804-3b22b323bff1(a)u25g2000prh.googlegroups.com>,
AI <vcpandya(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.
> >
> > > {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

OK, I found the Erdos 1939 paper.
The site http://www.renyi.hu/~p_erdos/Erdos4ht.html lists all
of Erdos' papers.
The one we want is
[1939*4] 1939-04 P. Erdos: Note on the product of consecutive integers,
II., J. London Math. Soc. 14 (1939), 245--249 MR1,39d; Zentralblatt
26,388.
in that catalogue.
The review in Math Reviews confirms that it contains a proof of
the result queried at the start of this thread.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: AI on 4 Dec 2009 08:04
On Nov 30, 3:43 am, Gerry Myerson <ge...(a)maths.mq.edi.ai.i2u4email>
wrote:
> In article
> <5a3fd17c-4f48-4245-8804-3b22b323b...(a)u25g2000prh.googlegroups.com>,
>
>
>
>
>
>  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.
>
> > > > {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
>
> OK, I found the Erdos 1939 paper.
> The sitehttp://www.renyi.hu/~p_erdos/Erdos4ht.htmllists all
> of Erdos' papers.
> The one we want is
> [1939*4] 1939-04 P. Erdos: Note on the product of consecutive integers,
> II., J. London Math. Soc. 14 (1939), 245--249 MR1,39d; Zentralblatt
> 26,388.
> in that catalogue.
> The review in Math Reviews confirms that it contains a proof of
> the result queried at the start of this thread.
>
> --
> Gerry Myerson (ge...(a)maths.mq.edi.ai) (i -> u for email)

Thanks for that link!
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