- Open Access
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.
- elliptic 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 .
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.
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