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How do I solve the Diophantine equation $ax^2 + bx + c = y^2$?

The approach I have so far is to use the transformation $$ X = 2ax + b, \\ Y = 2y. $$

Applying this, we get,

$$X^2 - dY^2 = n,$$

where $$n = b^2 - 4ac \text{ and } d = a.$$

$X^2 - dY^2 = n$ is a Pell equation which may be solved using the method for solving Pell-type equations.

Questions:

  1. Is there any other method?
  2. What is the complexity of the algorithm for finding the solution to the Pell equation?
vvg
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    Since, as you say, this is a slight transformation of a Pell equation (up to checking some parities), solving this is more or less equivalent to solving a Pell equation, and you can look up all sorts of literature about those. – Qiaochu Yuan Sep 18 '20 at 18:15
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    Modular method without continued fraction: link1, link2. Method by Norman Wildberger: link3. – Dmitry Ezhov Sep 18 '20 at 19:27
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    https://math.stackexchange.com/questions/128930/whenever-pells-equation-proof-is-solvable-it-has-infinitely-many-solutions/831363#831363 https://artofproblemsolving.com/community/c3046h1056199_general_pell_equation https://artofproblemsolving.com/community/c3046h1049910___4 https://artofproblemsolving.com/community/c3046h1048216__ – individ Sep 21 '20 at 04:39

2 Answers2

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The "usual" method is continued fractions. This is (asymptotically) slower than $O(p(n))$ for any polynomial $p$.

Using the quadratic sieve and assuming the generalized Riemann hypothesis (described in [L2002]), solutions can be found in time $\exp \left( O(\sqrt{\log n \log \log n}) \right)$, which is still (asymptotically) slower than any polynomial.

There are polynomial time quantum algorithms.[H2007]

[H2007]: Hallgren, Sean (2007), "Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem", Journal of the ACM, 54 (1): 1–19, doi:10.1145/1206035.1206039, S2CID 948064

[L2002]: Lenstra, H. W., Jr. (2002), "Solving the Pell Equation" (PDF), Notices of the American Mathematical Society, 49 (2): 182–192, MR 1875156

Eric Towers
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Lagrange worked extensively on this problem. His work is in French and hasn't been formally translated into English anywhere, although good artificial intelligence tools now exist in the form of Transkribus and DeepL. Here is a link to his voluminous work on these equations:

http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__2_377_0

Joebloggs
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