( 1/x + 1/y ) where x,y are positive integers

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henh...@gmail.com

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Aug 29, 2022, 12:10:00 AMAug 29
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is there a simple desc. for all the numbers that can be
expressed as ( 1/x + 1/y ) where x,y are positive integers ?

henh...@gmail.com

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Aug 29, 2022, 12:13:59 AMAug 29
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> is there a simple desc. for all the numbers that can be
> expressed as ( 1/x + 1/y ) where x,y are positive integers ?


Are there some rational number(s) z ( 0 < z < 2)
not expressible as ( 1/x + 1/y ) where x,y are positive integers ?

Eric Sosman

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Aug 29, 2022, 9:51:14 AMAug 29
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On 8/29/2022 12:09 AM, henh...@gmail.com wrote:
>
> is there a simple desc. for all the numbers that can be
> expressed as ( 1/x + 1/y ) where x,y are positive integers ?

"(0..2]" seems to cover it, if I haven't missed something.

--
eso...@comcast-dot-net.invalid
Look on my code, ye Hackers, and guffaw!

Mike Terry

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Aug 29, 2022, 10:53:13 AMAug 29
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There are obviously loads of rationals not expressible as 1/x + 1/y.

In fact, the set of numbers of that form has no limit points other than zero (which is not in the
set), so the set is discrete. [Every point in the set is isolated - it has a neighbourhood where
the point in question is the only number of the form 1/x + 1/y.]

Not sure what else to say ... um, every interval not containing zero contains only finitely many
numbers of the form 1/x + 1/y ...

Mike.

Eric Sosman

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Aug 29, 2022, 11:35:22 AMAug 29
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On 8/29/2022 9:51 AM, Eric Sosman wrote:
> On 8/29/2022 12:09 AM, henh...@gmail.com wrote:
>>
>> is there a simple desc. for all the numbers that can be
>> expressed as   ( 1/x + 1/y )  where x,y are positive integers ?
>
> "(0..2]" seems to cover it, if I haven't missed something.

Hmmm: Seems I missed something.

Jonathan Dushoff

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Aug 29, 2022, 12:35:18 PMAug 29
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On Monday, August 29, 2022 at 12:10:00 AM UTC-4, henh...@gmail.com wrote:
> is there a simple desc. for all the numbers that can be
> expressed as ( 1/x + 1/y ) where x,y are positive integers ?

On Monday, August 29, 2022 at 12:10:00 AM UTC-4, henh...@gmail.com wrote:
> is there a simple desc. for all the numbers that can be
> expressed as ( 1/x + 1/y ) where x,y are positive integers ?

Let k be the greatest common factor of x and y.

Then x=ka, y=kb, and 1/x + 1/y is (a+b)/(kab)

a+b is relatively prime to ab, but not necessarily to k. So this fraction in simplest form is ((a+b)/ℓ)/(mab), where ℓ is the gcf and k=ℓm.

Thus, fraction p/q is the sum of two unitary fractions exactly when we can find a, b relatively prime s.t. p|(a+b) and ab|q.

Jonathan

Jonathan Dushoff

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Aug 29, 2022, 5:04:01 PMAug 29
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For example, consider 7/2022 (since neither 8/2022 nor 9/2022 are in lowest terms).

We can choose a=6 and b=337 so that the product divides 2022 and 7 divides the sum. Then we need ℓ=(a+b)/p = 49, and we conclude that:

7/2022 = 1/294 + 1/16513

Jonathan

henh...@gmail.com

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Aug 29, 2022, 11:29:18 PMAug 29
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( i must've encountered this when (or before) i was 11 or 12,
but i can't remember it well )


thakns ! ---
that method (algorithm) allows one to determine which of the following are
expressible as ( 1/x + 1/y ) where x,y are positive integers :

3/101 , 4/101
3/102 , 4/102
3/103 , 4/103
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