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

From henhanna@gmail.com@21:1/5 to All on Sun Aug 28 21:13:58 2022
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 ?

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• From henhanna@gmail.com@21:1/5 to All on Sun Aug 28 21:09:58 2022
is there a simple desc. for all the numbers that can be
expressed as ( 1/x + 1/y ) where x,y are positive integers ?

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• From Eric Sosman@21:1/5 to henh...@gmail.com on Mon Aug 29 09:51:13 2022
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!

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• From Mike Terry@21:1/5 to henh...@gmail.com on Mon Aug 29 15:53:03 2022
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.

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• From Eric Sosman@21:1/5 to Eric Sosman on Mon Aug 29 11:35:19 2022
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!

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• From Jonathan Dushoff@21:1/5 to henh...@gmail.com on Mon Aug 29 09:35:16 2022
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

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• From Jonathan Dushoff@21:1/5 to Jonathan Dushoff on Mon Aug 29 14:04:00 2022
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

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• From henhanna@gmail.com@21:1/5 to Jonathan Dushoff on Mon Aug 29 20:29:16 2022
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

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