Generators and integer points on the elliptic curve y2=x3-nx

2013 ◽  
Vol 160 (4) ◽  
pp. 333-348 ◽  
Author(s):  
Yasutsugu Fujita ◽  
Nobuhiro Terai
Keyword(s):  
Elliptic Curve ◽  
Integer Points

scholarly journals The positive integral points on the elliptic curve 𝒚𝟐=𝟕𝒑𝒙(𝒙𝟐+𝟖)

2020 ◽  
Vol 165 ◽  
pp. 03046
Author(s):  
Du Xiancun ◽  
Jianhong Zhao ◽  
Lixing Yang
Keyword(s):  
Number Theory ◽  
Elliptic Curve ◽  
Integer Point ◽  
Integral Point ◽  
Analytic Number Theory ◽  
Elementary Number Theory ◽  
Integral Points ◽  
Integer Points ◽  
Elementary Methods ◽  
Special Case

The integral point of elliptic curve is a very important problem in both elementary number theory and analytic number theory. In recent years, scholars have paid great attention to solving the problem of positive integer points on elliptic curve 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), where 𝑘,𝑎,𝑏,𝑐 are integers. As a special case of 𝑦2 = 𝑘(𝑎𝑥2+𝑏𝑥+𝑐), when 𝑎 = 1,𝑏 = 0,𝑐 = 22𝑡−1, it turns into 𝑦2 = 𝑘𝑥(𝑥2+22𝑡−1), which is a very important case. However ,at present, there are only a few conclusions on it, and the conclusions mainly concentrated on the case of 𝑡 = 1,2,3,4. The case of 𝑡 = 1, main conclusions reference [1] to [7]. The case of 𝑡 = 2, main conclusions reference [8]. The case of 𝑡 = 3, main conclusions reference [9] to [11]. The case of 𝑡 = 4, main conclusions reference [12] and [13]. Up to now, there is no relevant result on the case of 𝑘 = 7𝑝 when 𝑡 = 2, here the elliptic curve is 𝑦2 = 7𝑝(𝑥2 + 8), this paper mainly discusses the positive integral points of it. And we obtained the conclusion of the positive integral points on the elliptic curve 𝑦2 = 7𝑝(𝑥2 + 8). By using congruence, Legendre symbol and other elementary methods, it is proved that the elliptic curve in the title has at most one integer point when 𝑝 ≡ 5,7(𝑚𝑜𝑑8).

ON THE NUMBER OF INTEGER POINTS ON THE ELLIPTIC CURVE y2 = x3 + Ax

2011 ◽  
Vol 07 (03) ◽  
pp. 611-621 ◽  
Author(s):  
KONSTANTINOS A. DRAZIOTIS
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Integer Points

It is given an upper bound for the number of the integer points of the elliptic curve y2 = x3 + Ax (A ∈ ℤ) and a conjecture of Schmidt is proven for this family of elliptic curves.

scholarly journals Integer Points and Independent Points on the Elliptic Curve $y^2=x^3-p^kx$

2011 ◽  
Vol 34 (2) ◽  
pp. 367-381 ◽  
Author(s):  
Yasutsugu FUJITA ◽  
Nobuhiro TERAI
Keyword(s):  
Elliptic Curve ◽  
Integer Points

scholarly journals Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials

2016 ◽  
Vol 68 (5) ◽  
pp. 1120-1158 ◽  
Author(s):  
Katherine E. Stange
Keyword(s):  
Elliptic Curve ◽  
Absolute Constant ◽  
Uniform Bound ◽  
Integral Points ◽  
Torsion Points ◽  
Integer Points ◽  
New Class ◽  
Singular Reduction ◽  
Division Polynomials ◽  
Explicit Formulae

AbstractAssuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/ℚ and non-torsion point P ∈ E(ℚ), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional, but not uniform, bound. The new ingredient is a collection of explicit formulæ for the sequence v(Ψn) of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on ĥ(P)/h(E) for integer points having two large integral multiples.

scholarly journals On the size of integer solutions of elliptic equations

1998 ◽  
Vol 57 (2) ◽  
pp. 199-206 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Elliptic Equations ◽  
Prime Ideals ◽  
Integer Points ◽  
Explicit Bound ◽  
Integer Solutions

We improve upon earlier effective bounds for the magnitude of integer points on an elliptic curve ε defined over a number field K. We slightly refine the dependence on the discriminant of K. In most of the previous papers, the estimates obtained are exponential in the height of ε. In this work, taking also into consideration the prime ideals dividing the discriminant of ε, we provide a totally explicit bound which is only polynomial in the height.

A NOTE ON SCHMIDT’S CONJECTURE

2017 ◽  
Vol 96 (2) ◽  
pp. 191-195
Author(s):  
DIMITRIOS POULAKIS
Keyword(s):  
Elliptic Curve ◽  
Algebraic Number ◽  
Number Fields ◽  
The Other ◽  
Algebraic Number Fields ◽  
Integer Points ◽  
Other Hand

Schmidt [‘Integer points on curves of genus 1’, Compos. Math. 81 (1992), 33–59] conjectured that the number of integer points on the elliptic curve defined by the equation $y^{2}=x^{3}+ax^{2}+bx+c$, with $a,b,c\in \mathbb{Z}$, is $O_{\unicode[STIX]{x1D716}}(\max \{1,|a|,|b|,|c|\}^{\unicode[STIX]{x1D716}})$ for any $\unicode[STIX]{x1D716}>0$. On the other hand, Duke [‘Bounds for arithmetic multiplicities’, Proc. Int. Congress Mathematicians, Vol. II (1998), 163–172] conjectured that the number of algebraic number fields of given degree and discriminant $D$ is $O_{\unicode[STIX]{x1D716}}(|D|^{\unicode[STIX]{x1D716}})$. In this note, we prove that Duke’s conjecture for quartic number fields implies Schmidt’s conjecture. We also give a short unconditional proof of Schmidt’s conjecture for the elliptic curve $y^{2}=x^{3}+ax$.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer

Implementation of an Elliptic Curve Scalar Multiplication Method Using Division Polynomials

2014 ◽  
Vol E97.A (1) ◽  
pp. 300-302 ◽  
Author(s):  
Naoki KANAYAMA ◽  
Yang LIU ◽  
Eiji OKAMOTO ◽  
Kazutaka SAITO ◽  
Tadanori TERUYA ◽  
...  
Keyword(s):  
Elliptic Curve ◽  
Scalar Multiplication ◽  
Division Polynomials ◽  
Elliptic Curve Scalar Multiplication

Enhanced Security Authentication of WiMAX using EC3A

2017 ◽  
Vol 7 (8) ◽  
pp. 219
Author(s):  
Mohd Javed ◽  
Khaleel Ahmad ◽  
Ahmad Talha Siddiqui
Keyword(s):  
Cellular Automata ◽  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Information Rate ◽  
Security Authentication ◽  
Encryption And Decryption ◽  
High Information

WiMAX is the innovation and upgradation of 802.16 benchmarks given by IEEE. It has numerous remarkable qualities, for example, high information rate, the nature of the service, versatility, security and portability putting it heads and shoulder over the current advancements like broadband link, DSL and remote systems. Though like its competitors the concern for security remains mandatory. Since the remote medium is accessible to call, the assailants can undoubtedly get into the system, making the powerless against the client. Many modern confirmations and encryption methods have been installed into WiMAX; however, regardless it opens with up different dangers. In this paper, we proposed Elliptic curve Cryptography based on Cellular Automata (EC3A) for encryption and decryption the message for improving the WiMAX security

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