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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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 r.e.s. Posts: 400 Registered: 12/4/04
Re: Integer pairs in sum of reciprocals
Posted: Jan 2, 2001 1:20 AM

"David Eppstein" wrote ...
| "r.e.s." 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 ?
| > (With c = 2001, I find 14 solutions.)
|
| You apparently forgot about the requirement that a<b.

yup (corrected in a separate posting)

| > I would be interested to know more about the number of
| > solutions as a function of c (e.g., asymtotically as c->oo).
|
| It's half the number of divisors of c^2.
| So, it doesn't behave very well asymptotically.

When I had earlier plotted the # solutions for 1<=c<=2001,
there seemed to be an envelope shaping up that suggested an
upper bound that might be very roughly logarithmic. Is the
"number of divisors" function, say d(c^2), known to behave
anything like that?

--r.e.s.