Linear Diophantine EquationDate: 02/05/2003 at 11:00:20 From: Jane Subject: Linear Diophantine equation Given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, b^4|a^5,....., prove that a = b. I do not know how to approach this question. I thought about contradiction or induction but I'm not too familiar with either method. Date: 02/06/2003 at 09:56:16 From: Doctor Jacques Subject: Re: Linear Diophantine equation Hi Jane, If both a and b are equal to 1, we certainly have a=b. We may therefore assume that at least one of {a,b} is not equal to 1. Let us assume that a <> b. If a<>b, their prime factorisations are different. It means that we can find a prime p such that p^j | a, p^i | b, i and j are the greatest such exponents, and i <> j. Let us assume, for example, that i < j (the other case is similar). Note that this includes the case i = 0. The given relations imply (we use only every other relation): j <= 2i 3j <= 4i 5j <= 6i .... (2n - 1) j <= 2n*i for all n. If we divide the last equation by 2n, we find: j - (j/2n) <= i As this relation must be true for all n, we may choose n > j. The inequation then becomes: j - (1/2) < i On the other hand, we assumed that i < j Now, remember that i and j are integers. This should allow you to find a contradiction. Please feel free to write back if you are still stuck. - Doctor Jacques, The Math Forum http://mathforum.org/dr.math/ |
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