Are x, y, z pairwise coprime or coprime together?
In mathematics, the case of variable names matters. For example, X and x are different variables.
Okay, so I'm stuck on this for an hour now, so I'd really appreciate some help, please:
X, Y, Z are coprime integers. We also know that 1/x + 1/y = 1/z. Prove that (X+Y) is a square number.
This should be entry level university math, but I'm stuck with it. I hope I posted to the right topic. Thanks for the help.
OK, here is a sketch. Suppose that p, q, r are prime, i, j₁, j₂ and k are positive integers, and , . In the general case, the variables here should be vectors. Without loss of generality, let . Then
Let . Then the only primes that could divide s are p, q and r. If p | s, then , a contradiction. Similarly, r | s is false. Thus for some j. Moreover, ; otherwise q ∈ GCD(x, y, z). Finally, if , then q | s implies , a contradiction. Therefore, , and .
See if you can shorten this and if it can be extended to the general case.
Generally this equation quotient more general equation.
Solutions in integers can be written by expanding the number of factorization: $N=ab$
And vospolzovavschis solutions of Pell's equation:
- what some integer number given by us.
Solutions can be written:
Perhaps these formulas for someone too complicated. Then equation:
If we ask what ever number: That the following sum can always be factored:
Solutions can be written.