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  • 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 ?

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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 ?

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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!

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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!

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • 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

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)