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Topic: Integer pairs in sum of reciprocals
Replies: 39   Last Post: Jan 22, 2001 6:02 PM

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355113@my-deja.com

Posts: 19
Registered: 12/13/04
Re: Integer pairs in sum of reciprocals
Posted: Jan 11, 2001 12:08 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply



In article <3A5139E4.94731FAA@hot.rr.com>,
"Ross A. Finlayson" <rfinlayson@hot.rr.com> wrote:
>
> saxon970@yahoo.com wrote:

> >
> > How many pairs of positive integers a and b are there such that
> > a < b and 1/a + 1/b = 1/2001 ?

>
> Set a to 2001 or greater and b to a large value. The result is
> very close to 1/2001, where 2001 is called c,
>
> 1/a + 1/b = 1/c, or
>
> a ^-1 + b ^-1 = c^-1
>
> There are probably infinite solutions.


saxon's message has disappeared from Deja, but as several
people mentioned (including Nathan until he changed his mind!)
there are 13 solutions:

Writing 1/a + 1/b = 1/2001 as u.v = 2001^2 = 3^2.23^2.29^2,
with u, v = a - 2001, b - 2001, there are 3^3 = 27 integer
pairs |u|, |v|.

(Each 3 here is for the 3 possible powers 0, 1, 2 of each
prime independently that can divide say u. More generally,
if the number was p^m.q^n.. with p, q,.. prime then the
number of pairs would be (m+1).(n+1)..).

If u, v < 0 then for 0 < a, b we require |u|, |v| < 2001
which is impossible. Thus 0 < u, v.

There are thus 13 such pairs with 0 < u < v, i.e. half
of the 26 pairs with u not equal v.


Cheers

John_Ramsden@acer.de


Sent via Deja.com
http://www.deja.com/






Date Subject Author
1/1/01
Read Integer pairs in sum of reciprocals
saxon970@yahoo.com
1/1/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Jan Kristian Haugland
1/2/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Jan Kristian Haugland
1/2/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Steve Lord
1/2/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Ross A. Finlayson
1/2/01
Read Re: Integer pairs in sum of reciprocals
Dik T. Winter
1/3/01
Read Re: Integer pairs in sum of reciprocals
Philip Anderson
1/11/01
Read Re: Integer pairs in sum of reciprocals
355113@my-deja.com
1/1/01
Read Re: Integer pairs in sum of reciprocals
David Eppstein
1/2/01
Read Re: Integer pairs in sum of reciprocals
r.e.s.
1/2/01
Read Re: Integer pairs in sum of reciprocals
David Eppstein
1/2/01
Read Re: Integer pairs in sum of reciprocals
r.e.s.
1/2/01
Read Re: Integer pairs in sum of reciprocals
r.e.s.
1/5/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/5/01
Read Re: Integer pairs in sum of reciprocals
r.e.s.
1/5/01
Read Re: Integer pairs in sum of reciprocals
Randy Poe
1/6/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/6/01
Read Re: Integer pairs in sum of reciprocals
David C. Ullrich
1/9/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/9/01
Read Re: Integer pairs in sum of reciprocals
David C. Ullrich
1/11/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/12/01
Read Re: Integer pairs in sum of reciprocals
David C. Ullrich
1/13/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/13/01
Read Re: Integer pairs in sum of reciprocals
David C. Ullrich
1/18/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/18/01
Read Re: Integer pairs in sum of reciprocals
David C. Ullrich
1/22/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/4/01
Read Re: Integer pairs in sum of reciprocals
Brian Evans
1/4/01
Read Re: Integer pairs in sum of reciprocals
Jan Kristian Haugland
1/5/01
Read Re: Integer pairs in sum of reciprocals
the_great_nathan@my-deja.com
1/5/01
Read Re: Integer pairs in sum of reciprocals
Dave Seaman
1/5/01
Read Re: Integer pairs in sum of reciprocals
David Eppstein
1/5/01
Read Re: Integer pairs in sum of reciprocals
Jan Kristian Haugland
1/6/01
Read Re: Integer pairs in sum of reciprocals
Damiano Scapeccia
1/7/01
Read Re: Integer pairs in sum of reciprocals
Adam Stephanides

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