MATH437 December 2006
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Suppose that x,y,z are positive integers satisfying and . Show that there are integers u and v for which and .
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The Fundamental Theorem of Arithmetic is the major idea in this proof. Start with the term and proceed from there.
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By the fundamental theorem of arithmetic, let be the prime factorization. Suppose that for any i. Then since x and y are coprime, this means that . Thus, we must have that .
If one of x or y (or both) has no prime factors, then since these numbers are positive, they must be equal to 1 which is a square.
Repeating this argument for all such i and noticing that this argument is symmetric in x and y, we see that
completing the proof.
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