scholarly journals On Arithmetic Progressions on Elliptic Curves

1999 ◽  
Vol 8 (4) ◽  
pp. 409-413 ◽  
Author(s):  
Andrew Bremner
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progressions

scholarly journals INFINITUDE OF ELLIPTIC CARMICHAEL NUMBERS

2012 ◽  
Vol 92 (1) ◽  
pp. 45-60 ◽  
Author(s):  
AARON EKSTROM ◽  
CARL POMERANCE ◽  
DINESH S. THAKUR
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Residue Class ◽  
Complex Multiplication ◽  
Weak Form ◽  
Arithmetic Progressions ◽  
Carmichael Numbers ◽  
The Third ◽  
Fermat’S Little Theorem

AbstractIn 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard conjecture on the bound on the least prime in (special) arithmetic progressions. Our results are somewhat more general than both the 1999 dissertation of the first author (written under the direction of the third author) and a 2010 paper on Carmichael numbers in a residue class written by Banks and the second author.

On the rank of elliptic curves with long arithmetic progressions

2017 ◽  
Vol 148 (1) ◽  
pp. 47-68
Author(s):  
Dustin Moody ◽  
Arman Shamsi Zargar
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progressions

scholarly journals Heron quadrilaterals with sides in arithmetic or geometric progression

1999 ◽  
Vol 59 (2) ◽  
pp. 263-269 ◽  
Author(s):  
R.H. Buchholz ◽  
J.A. MacDougall
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
Infinite Family ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Geometric Progression ◽  
Complete Characterisation

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterisation is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose sides form either a geometric or an arithmetic progression. The solution of both quadrilateral cases involves searching for rational points on certain elliptic curves.

scholarly journals Elliptic Curves with Long Arithmetic Progressions Have Large Rank

2020 ◽  
Author(s):  
Natalia Garcia-Fritz ◽  
Hector Pasten
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
Nevanlinna Theory ◽  
Uniform Boundedness ◽  
Rational Numbers ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Large Rank ◽  
Arithmetic Statistics

Abstract For any family of elliptic curves over the rational numbers with fixed $j$-invariant, we prove that the existence of a long sequence of rational points whose $x$-coordinates form a nontrivial arithmetic progression implies that the Mordell–Weil rank is large, and similarly for $y$-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond’s quantitative extension of results of Faltings.

scholarly journals Quadratic sequences of powers and Mohanty’s conjecture

2018 ◽  
Vol 14 (02) ◽  
pp. 479-507 ◽  
Author(s):  
Natalia Garcia-Fritz
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progression ◽  
General Type ◽  
Function Fields ◽  
Hyperelliptic Curves ◽  
Arithmetic Progressions ◽  
Surfaces Of General Type ◽  
Genus Zero ◽  
Curves Of Low Genus

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

scholarly journals On Simultaneous Arithmetic Progressions on Elliptic Curves

2006 ◽  
Vol 15 (4) ◽  
pp. 471-478 ◽  
Author(s):  
Irene García-Selfa ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Arithmetic Progressions

scholarly journals Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t]

2019 ◽  
Vol 16 (05) ◽  
pp. 1013-1030
Author(s):  
Edva Roditty-Gershon ◽  
Chris Hall ◽  
Jonathan P. Keating
Keyword(s):  
Elliptic Curves ◽  
Conjugacy Classes ◽  
Arithmetic Progressions ◽  
Divisor Function ◽  
Legendre Curve ◽  
Divisor Functions ◽  
Matrix Integrals ◽  
Usual Formula ◽  
Special Case

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.

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