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I was reading a pdf (Arithmetic Progressions of Three Squares by Keith Conrad) and I have a question about Theorem 3.5. It says that we can use Dirichlet's theorem on primes in arithmetic progression to prove the Theorem, but the proof is on pages 44-45 and I can't find it. Does anyone know the proof with Dirichlet's theorem or have the pages 44-45 from Arithmetic Progressions of Three Squares ?

"Theorem 3.5. For rational $ n\neq 0 $, the only nonidentity rational points on $ y^2=x^3-n^2x $ of finite order are $ (0,0), (n,0), and (-n,0) $. "

Dr.Mathematics
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    It is about the rational points on the torsion, not on the whole curve. Search 'Koblitz nozdr' referenced by https://kconrad.math.uconn.edu/blurbs/ugradnumthy/3squarearithprog.pdf which shows $E : y^2=x^3-n^2 x$ has $#E(\mathbb{F}p)=p+1$ when $p \equiv 3 \bmod 4$, together with the morphism $E(\mathbb{Q}){tors} \to E(\mathbb{F}_p)$ and Dirichlet's theorem it should do it – reuns Jan 26 '19 at 18:51
  • Thanks. Furthermore, I have a little problem with Corollary 3.3. I can't understand how we can pass between $ nk^2= m^3 - m $ and $y^2= x^3 - n^2x $. – Dr.Mathematics Jan 26 '19 at 19:50
  • It is explained juste below $x = nm, y = m^2 k$ – reuns Jan 26 '19 at 21:13

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