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