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Fix a square-free integer $d > 1$, and assume that Z[√d] satisfies the Fundamental Theorem of Arithmetic. Show that the equation $y^2 = x^3 + d$ has only finitely many integral solutions.

Reference: An introduction to number theory. Graham Everest. Exercise 2.15

Bernard
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1 Answers1

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The solution follows from the material in this section easily, imitating the proof of Theorem 2.11, and Theorem 2.12. Also Siegel's Theorem - Theorem 2.13 is in the book. This solves such problems in general. In fact, Carl Ludwig Siegel proved that a non-singular plane cubic equation has only finitely many integral solutions. References on this site:

Integral points on an elliptic curve

Effective proofs of Siegel's theorem using arithmetic geometry

Furthermore there is Terry Tao's blog.

Dietrich Burde
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  • I tried to prove that y+√d,y-√d are co-primes, by using the gcd of them, and so come to prove that x^3 is odd, by assuming it is even and taking mod 8. Then y^2=d mod 8 and so because of the square, d must be either 0 or 1 or 4. When testing 0 or 4 I came to contradiction with d a square free, but when I come to test d = 1 mod 8, I don’t know how to come to a contradiction. –  Oct 29 '17 at 14:32