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...)?
Since I doublepressed, I might as well insert something different here.
If we are not concerned about the GCD condition then we can extend xdog's case to those where x is a square, and k can be anything.
Then the triplet c=x^2(k1), b=x^2k, a=k(x^2(k1)), will produce a+b=(kx)^2, but there are other cases as well:
x k a b c a+b sqrt
1 9 72 9 8 81 9
9 1 0 9 0 9 3
2 9 144 18 16 162
9 2 18 18 9 36 6
3 9 216 27 24 243
9 3 54 27 18 81 9
4 9 288 36 32 324 18
9 4 108 36 27 144 12
5 9 360 45 40 405
9 5 180 45 36 225 15
6 9 432 54 48 486
9 6 270 54 45 324 18
7 9 504 63 56 567
9 7 378 63 54 441 21
8 9 576 72 64 648
9 8 504 72 63 576 24
9 9 648 81 72 729 27
Edited on March 7, 2020, 11:22 pm

Posted by broll
on 20200307 22:46:30 