Primitive Harmonic Triplets

Define a "primitive harmonic triplet" as any ordered triplet (x,y,z),(x,y,z), where x,y,zZ+x,y,z\in\mathbb{Z}^+ and gcd(x,y,z)=1,\gcd{(x,y,z)}=1, such that 1x+1y=1z.\frac{1}{x}+\frac{1}{y}=\frac{1}{z}. Are there infinitely many primitive harmonic triplets?

  • If the answer is no, enter the number of ordered pairs (x,y)(x,y) such that (x,y,6)(x,y,6) is a primitive harmonic triplet.
  • If the answer is yes, enter the number of ordered pairs (x,y)(x,y) such that (x, y, 25×34×53×72×11)\big(x,\ y,\ 2^{5}\times 3^{4}\times 5^{3}\times 7^{2}\times 11\big) is a primitive harmonic triplet.

If you enjoyed this problem, you might want to check this one and this one out, too.

×

Problem Loading...

Note Loading...

Set Loading...