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co-prime unit fractions (Posted on 2020-03-06) Difficulty: 3 of 5
There are 3 positive integers a, b, c such that 1/c=1/a+1/b. If the greatest common divisor of a, b, c is 1, then what type of number must a+b be(e.g square number, cube number, triangular number...)?

No Solution Yet Submitted by Danish Ahmed Khan    
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re: Possible solution, first posting | Comment 5 of 7 |
(In reply to Possible solution by broll)

"So a+b . . . can only be square with a GCD of 1 if c=(k-1), b=k, and a=k(k-1))"


I believe (without proof) that's true for prime c but it fails if c is composite.  For example, (a,b,c)=(14,35,10) or (26,143,22) or (114,1330,105) are solutions with GCD=1 but with a different form.

  Posted by xdog on 2020-03-08 12:38:48
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