Integral points on the elliptic curve
© Karaatlıand Keskin; licensee Springer 2013
Received: 9 January 2013
Accepted: 6 March 2013
Published: 2 May 2013
We give a new proof that the elliptic curve has only the integral points and using elementary number theory methods and some properties of generalized Fibonacci and Lucas sequences.
Keywordselliptic curves integral point generalized Fibonacci and Lucas sequences
is . Then the same problem was dealt with by some authors. In , Zhu and Chen found all integral points on (1.3) by using algebraic number theory and p-adic analysis. In , Wu proved that (1.3) has only the integral points and using some results of quartic Diophantine equations with elementary number methods. After that, in , the authors found the integral points on (1.3) using similar methods to those given in . In this paper, we determine that the largest integral point on the elliptic curve is by using elementary number theory methods and some properties of generalized Fibonacci and Lucas sequences. Our proof is extremely different from the proofs of the others.
In this section, we present two theorems and some well-known identities regarding the sequences and , which will be useful during the proof of the main theorem.
We state the following theorem from .
Theorem 2.1 Let . If with and , then or .
The following theorem is a well-known theorem (see ).
Theorem 2.2 Let and . Then .
3 Proof of the main theorem
The main theorem we deal with here is as follows.
Theorem 3.1 The elliptic curve has only the integral points and .
for some positive integers a and b.
If , then from (3.2) we get . Completing the square gives . This implies that . It can be easily shown that there are no integers a and b satisfying the previous equation.
Working on modulo 8 shows that (3.3) is impossible.
Working on modulo 8 shows that (3.4) is impossible.
So, working on modulo 8 and using (3.11) in Eq. (3.9) lead to a contradiction.
Since and , a simple computation shows that . Moreover, since and , we get and therefore . Substituting into gives . Hence, the theorem is proved, the elliptic curve has only the integral points and , which is the largest integral point on it. This completes the proof of the main theorem. □
Dedicated to Professor Hari M Srivastava.
- Kalman D, Mena R: The Fibonacci numbers-exposed. Math. Mag. 2003, 76: 167–181.MathSciNetView ArticleGoogle Scholar
- Karaatlı O, Keskin R: On some diophantine equations related to square triangular and balancing numbers. J. Algebra, Number Theory: Adv. Appl. 2010, 4(2):71–89.Google Scholar
- Muskat JB: Generalized Fibonacci and Lucas sequences and rootfinding methods. Math. Comput. 1993, 61: 365–372.MathSciNetView ArticleGoogle Scholar
- Rabinowitz S: Algorithmic manipulation of Fibonacci identities. 6. In Application of Fibonacci Numbers. Kluwer Academic, Dordrecht; 1996:389–408.View ArticleGoogle Scholar
- Ribenboim P: My Numbers, My Friends. Springer, New York; 2000.Google Scholar
- Baker A:The Diophantine equation . J. Lond. Math. Soc. 1968, 43: 1–9.View ArticleGoogle Scholar
- Stroeker RJ, Tzanakis N: On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement. Exp. Math. 1999, 8: 135–149.MathSciNetView ArticleGoogle Scholar
- Stroeker RJ, Tzanakis N: Computing all integer solutions of a genus 1 equation. Math. Comput. 2003, 72: 1917–1933.MathSciNetView ArticleGoogle Scholar
- Zagier D: Large integral points on elliptic curves. Math. Comput. 1987, 48: 425–436.MathSciNetView ArticleGoogle Scholar
- Zhu H, Chen J:Integral points on . J. Math. Study 2009, 42(2):117–125.MathSciNetGoogle Scholar
- Wu H:Points on the elliptic curve . Acta Math. Sin., Chin. Ser. 2010, 53(1):205–208.Google Scholar
- He Y, Zhang W: An elliptic curve having large integral points. Czechoslov. Math. J. 2010, 60(135):1101–1107.View ArticleGoogle Scholar
- Mignotte M, Pethő A: Sur les carrés dans certanies suites de Lucas. J. Théor. Nr. Bordx. 1993, 5(2):333–341.View ArticleGoogle Scholar
- Ribenboim P: An algorithm to determine the points with integral coordinates in certain elliptic curves. J. Number Theory 1999, 74: 19–38.MathSciNetView ArticleGoogle Scholar
- Nagell T: Introduction to Number Theory. Wiley, New York; 1981.Google Scholar
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