An Algorithm to Determine the Points with Integral Coordinates in Certain Elliptic Curves

Ribenboim, Paulo

doi:10.1006/jnth.1998.2290

Cited by 10 publications

(7 citation statements)

References 15 publications

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“…Put n 0 m = n (then, of course, U n 0 | U n ); also, put d = gcd(U n 0 , U n /U n 0 ). By results of Lehmer [8] (see also (2.2) of [15]), d | m ( 1 ), hence P(d) ⊆ P(m) ⊆ S. On the other hand, the relation U n = k2 can be written…”

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confidence: 95%

“…We will often use various properties of the Lucas sequences, which can be found, for example, in the first two sections of [15]. For any non-zero integer x we define…”

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confidence: 99%

“…( 1 ) The assumption P > 0 made in [15] does not affect the result, as U n (−P, Q) and U n (P, Q) differ at most in sign.…”

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confidence: 99%

“…Terms of Lucas sequences being almost squares. In this section we will show that for fixed non-zero integer k and parameters P, Q which satisfy (2), the problem of finding all n for which (15) U n (P, Q) = k2 leads to finitely many quartic Thue-Mahler equations, where the subscript n will appear in the exponents of the prime numbers on the right-hand side of these equations. Thue-Mahler equations, in general, can be explicitly solved, at least in principle, by the method explained in great detail in [21].…”

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confidence: 99%

“…It is of some interest to note at this point that for our special purposes we do not need the complete solution of the Thue-Mahler equations but only a small upper bound for the unknown exponents of the right-hand side; we make a short discussion of this issue at the beginning of Section 6. By Section 1 of [15] we have…”

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Lucas sequences whose nth term is a square or an almost square

Bremner

Tzanakis

2007

Acta Arith.

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Lucas sequences whose nth term is a square or an almost square by A. Bremner (Tempe, AZ) and N. Tzanakis (Iraklion)1. Introduction. Let P and Q be non-zero integers. The Lucas sequence {U n (P, Q)} is defined byHistorically, there has been much interest in when the terms of such sequences are perfect squares (or higher powers), and we summarize here the numerous and diverse results. Ljunggren [9] shows that if (P, Q) = (2, −1) and n ≥ 2, then U n is a perfect square precisely for U 7 = 13 2 , and U n = 22 precisely for U 2 = 2. The sequence {U n (1, −1)} is the familiar Fibonacci sequence, and Cohn [5] proved in 1964 that the only perfect square greater than 1 in this sequence is U 12 = 12 2 . Ribenboim and McDaniel [17] show that if P is even and Q ≡ 1 mod 4, then U n (P, Q) = 2 imposes necessary conditions on the prime factorization of n. Earlier, in [16], the same authors show with only elementary methods that when P and Q are odd, and P 2 − 4Q > 0, then U n = 2 only for n = 0, 1, 2, 3, 6 or 12; and that there are at most two indices greater than 1 for which U n can be square. They characterize fully the instances when U n = 2, for n = 2, 3, 6. Bremner & Tzanakis [2] extend these results by determining all Lucas sequences {U n (P, Q)} with U 12 = 2, subject only to the restriction that gcd(P, Q) = 1 (it turns out that the Fibonacci sequence provides the only example). Under the same hypothesis, all Lucas sequences {U n (P, Q)} with U 9 = 2 are determined. In a later paper, the same authors [3] show that if n = 2, . . . , 7 then U n (P, Q) is square for infinitely many coprime P, Q and determine all sequences {U n (P, Q)} with U n (P, Q) = 2, n = 8, 10, 11. We discuss in this paper the more general problem of finding all integers n, P , Q, for which U n (P, Q) = k2 for a given integer k. Results of Pethő [14], Shorey and Stewart [19], and Shorey and Tijdeman [20], show the finiteness of the number of solutions of U n (P, Q) = k2 for fixed P, Q.

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confidence: 95%

“…We will often use various properties of the Lucas sequences, which can be found, for example, in the first two sections of [15]. For any non-zero integer x we define…”

mentioning

confidence: 99%

“…( 1 ) The assumption P > 0 made in [15] does not affect the result, as U n (−P, Q) and U n (P, Q) differ at most in sign.…”

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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