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I'm reading Silverman/Tate's Rational Points on Elliptic Curves and pg 15 states:

(1) $\quad aX^2 + bY^2 = cZ^2$ (to be solved in integers)

"Legendre's theorem states that there is an integer m, depending in a simple fashion on a, b, c, so that the above equation (1) has a solution in integers, not all zero, if and only if the congruence:

$$ aX^2 + bY^2 \equiv cZ^2 \mod m$$

has a solution in integers relatively prime to $m$."

I'm guessing that $m$ is the product of $a,b,c$ but I want to make sure.

Michael Hardy
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Ben Kushigian
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    I don't think the phrase "depending in a simple fashion" means anything technical here, it is just avoid writing out the full result. Certainly, there is no evidence in this language that he is using something called "simple dependence." If he doesn't define the term prior in the book, then that makes it even more likely... – Thomas Andrews Dec 27 '14 at 02:21
  • I do not see why $m=abc$ would suffice. – Will Jagy Dec 27 '14 at 02:30
  • Thanks, I'll change the title and throw some back story in here. – Ben Kushigian Dec 27 '14 at 02:37
  • Suppose I made sure that there are solutions. How do You write a formula which would give the solution to this equation? – individ Dec 27 '14 at 06:20
  • I assume Legendre's theorem is defined in that book? – Thomas Andrews Dec 27 '14 at 15:03
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    Possible the theorem listed here? http://math.stackexchange.com/questions/27471/proof-of-legendres-theorem-on-the-ternary-quadratic-form – Thomas Andrews Dec 27 '14 at 18:01
  • Thanks Thomas - that is the theorem. The author doesn't actually list the theorem in the text (other than what I included up top) – Ben Kushigian Jan 19 '15 at 16:54

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