Quadratic Number Fields and Integer Solutions
Date: 08/16/2007 at 08:01:19 From: ziya Subject: my question is about the number theory Prove that the equation 34*y^2 - x^2 = 1 in Z (integer number set) has no answer. I have tested different module but it has no result.
Date: 08/16/2007 at 21:19:01 From: Doctor Vogler Subject: Re: my question is about the number theory Hi Ziya, Thanks for writing to Dr. Math. You can't prove that it doesn't have integer solutions using modular arithmetic since it DOES have rational solutions. For example, 34(1/3)^2 - (5/3)^2 = 1, and 34(1/5)^2 - (3/5)^2 = 1. Therefore, mod p^k for any prime p, you can invert either 3 or 5 (or usually both) and get solutions mod p^k. Then the Chinese Remainder Theorem can be used to get solutions mod m for ANY positive integer m. The only way I can think of to prove that your equation has no integer solutions is to use the theory of quadratic number fields. But you can still prove that it has no integer solutions using some algebraic number theory. Someone familiar with number fields would write your equation as x^2 - 34 y^2 = -1 and would factor it in the the ring of algebraic integers of the form a + b*sqrt(34) as (x - y*sqrt(34))(x + y*sqrt(34)) = -1, or Norm(x + y*sqrt(34)) = -1. Such a solution would give two "units" in this ring, and there is a theorem (Dirichlet's Unit Theorem) which says that all units in this ring are powers of a single number (called the "Fundamental Unit") possibly times -1. There are also algorithms for computing the fundamental unit of a ring like Z[sqrt(N)], and for your ring it turns out to be 35 + 6*sqrt(34). When the norm of the fundamental unit is -1, its square has norm +1, and half of the units will give a solution to x^2 - N y^2 = -1 and the other half will give a solution to x^2 - N y^2 = 1. But 35 + 6*sqrt(34) has norm +1 (not -1), and when this happens, all units give solutions to the equation x^2 - N y^2 = 1, which means that you can never get -1 on the right. There are plenty of N's for which this is true, and there are some theorems stating that certain kinds of N will always have fundamental units of a particular norm, but there are other kinds of N where you just have to compute the fundamental unit and calculate its norm. Algebraic number theory (and number fields) is a fascinating subject with a lot of very nice theorems. If you are interested in the subject, you could start by reading "Number Fields" by Marcus, although there are plenty of other good books on the subject as well. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/
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