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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.

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    $\begingroup$ 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... $\endgroup$
    – Thomas Andrews
    Dec 27, 2014 at 2:21
  • $\begingroup$ I do not see why $m=abc$ would suffice. $\endgroup$
    – Will Jagy
    Dec 27, 2014 at 2:30
  • $\begingroup$ Thanks, I'll change the title and throw some back story in here. $\endgroup$
    – Ben Kushigian
    Dec 27, 2014 at 2:37
  • $\begingroup$ Suppose I made sure that there are solutions. How do You write a formula which would give the solution to this equation? $\endgroup$
    – individ
    Dec 27, 2014 at 6:20
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    $\begingroup$ Possible the theorem listed here? math.stackexchange.com/questions/27471/… $\endgroup$
    – Thomas Andrews
    Dec 27, 2014 at 18:01

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