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