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.
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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.
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.
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
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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.
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:
And more:
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.