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interests / rec.puzzles / Re: ( 1/x + 1/y ) where x,y are positive integers

SubjectAuthor
* ( 1/x + 1/y ) where x,y are positive integershenh...@gmail.com
+* Re: ( 1/x + 1/y ) where x,y are positive integershenh...@gmail.com
|`- Re: ( 1/x + 1/y ) where x,y are positive integersMike Terry
+* Re: ( 1/x + 1/y ) where x,y are positive integersEric Sosman
|`- Re: ( 1/x + 1/y ) where x,y are positive integersEric Sosman
`* Re: ( 1/x + 1/y ) where x,y are positive integersJonathan Dushoff
 `* Re: ( 1/x + 1/y ) where x,y are positive integersJonathan Dushoff
  `- Re: ( 1/x + 1/y ) where x,y are positive integershenh...@gmail.com

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

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Subject: ( 1/x + 1/y ) where x,y are positive integers
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Mon, 29 Aug 2022 04:09 UTC

is there a simple desc. for all the numbers that can be
expressed as ( 1/x + 1/y ) where x,y are positive integers ?

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

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Subject: Re: ( 1/x + 1/y ) where x,y are positive integers
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Mon, 29 Aug 2022 04:13 UTC

> 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 ?

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

<teig8f$13gsb$1@dont-email.me>

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From: esos...@comcast-dot-net.invalid (Eric Sosman)
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Subject: Re: ( 1/x + 1/y ) where x,y are positive integers
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 by: Eric Sosman - Mon, 29 Aug 2022 13:51 UTC

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.

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

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

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From: news.dea...@darjeeling.plus.com (Mike Terry)
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 by: Mike Terry - Mon, 29 Aug 2022 14:53 UTC

On 29/08/2022 05:13, 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 ?
>
>
> Are there some rational number(s) z ( 0 < z < 2)
> not expressible as ( 1/x + 1/y ) where x,y are positive integers ?
>

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.

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

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 by: Eric Sosman - Mon, 29 Aug 2022 15:35 UTC

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.

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

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

<956b7a6f-7957-486e-929f-1d54aefaece8n@googlegroups.com>

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Subject: Re: ( 1/x + 1/y ) where x,y are positive integers
From: jdush...@gmail.com (Jonathan Dushoff)
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 by: Jonathan Dushoff - Mon, 29 Aug 2022 16:35 UTC

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

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

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Subject: Re: ( 1/x + 1/y ) where x,y are positive integers
From: jdush...@gmail.com (Jonathan Dushoff)
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 by: Jonathan Dushoff - Mon, 29 Aug 2022 21:04 UTC

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

On Monday, August 29, 2022 at 12:35:18 PM UTC-4, Jonathan Dushoff wrote:
> 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

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

<9116ae21-de31-4d07-9940-6e1814c0b75an@googlegroups.com>

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Subject: Re: ( 1/x + 1/y ) where x,y are positive integers
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Tue, 30 Aug 2022 03:29 UTC

On Monday, August 29, 2022 at 2:04:01 PM UTC-7, Jonathan Dushoff wrote:
> 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
> On Monday, August 29, 2022 at 12:35:18 PM UTC-4, Jonathan Dushoff wrote:
> > 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

( 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

1
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