Principal forms X 2+nY 2 representing many integers

Brink, D.M.; Moree, Pieter; Osburn, Robert

doi:10.1007/s12188-011-0059-y

Cited by 6 publications

(4 citation statements)

References 15 publications

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“…. = 0, thus establishing the falsity of Ramanujan's claim (3). Since γ S B < 1/2, it follows by Corollary 1 that actually the Ramanujan approximation is better.…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 78%

“…In the special case where f = X 2 + nY 2 , a remark in a paper of Shanks seemed to suggest that he thought C f would be maximal in case n = 2. However, the maximum does not occur for n = 2, see Brink et al [3].…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 97%

“…Since his work was not published in a mathematical journal it got forgotten and later partially rediscovered by mathematicians such as James and Pall. For a brief description of the proof approach of Bernays see Brink et al [3]. We like to point out that in general the estimate…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 99%

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Counting numbers in multiplicative sets: Landau versus Ramanujan

Moree

2011

Preprint

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A set S of integers is said to be multiplicative if for every pair m and n of coprime integers we have that mn is in S iff both m and n are in S. Both Landau and Ramanujan gave approximations to S(x), the number of n ≤ x that are in S, for specific choices of S. The asymptotical precision of their respective approaches are being compared and related to Euler-Kronecker constants, a generalization of Euler's constant γ = 0.57721566 . . .. This paper claims little originality, its aim is to give a survey on the literature related to this theme with an emphasis on the contributions of the author (and his coauthors).

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“…. = 0, thus establishing the falsity of Ramanujan's claim (3). Since γ S B < 1/2, it follows by Corollary 1 that actually the Ramanujan approximation is better.…”

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 78%

Section: Some Euler-kronecker Constants Related To Binary Quadratic F...mentioning

confidence: 97%

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Counting numbers in multiplicative sets: Landau versus Ramanujan

Moree

2011

Preprint

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“…Theorem 2.1. (Landau-Ramanujan [5,7,34]) The number of positive integers smaller than n that are the sum of two squares is Θ(n/ √ log n).…”

Section: The Structure Of Point Sets With Few Distinct Distancesmentioning

confidence: 99%

Distinct Distances: Open Problems and Current Bounds

Sheffer

2014

Preprint

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Finding orientations of supersingular elliptic curves and quaternion orders

Arpin,

Clements,

Dartois

et al. 2024

Des. Codes Cryptogr.

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An oriented supersingular elliptic curve is a curve which is enhanced with the information of an endomorphism. Computing the full endomorphism ring of a supersingular elliptic curve is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is $$\mathfrak {O}$$ O -orientable for a fixed imaginary quadratic order $$\mathfrak {O}$$ O provides non-trivial information towards computing an endomorphism corresponding to the $$\mathfrak {O}$$ O -orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at p and $$\infty $$ ∞ . We provide code implementations in Sagemath (in Stein et al. Sage Mathematics Software (Version 10.0), The Sage Development Team, http://www.sagemath.org, 2023) which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to O(p), even for cryptographically sized p.

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Principal forms X 2+nY 2 representing many integers

Cited by 6 publications

References 15 publications

Counting numbers in multiplicative sets: Landau versus Ramanujan

Counting numbers in multiplicative sets: Landau versus Ramanujan

Distinct Distances: Open Problems and Current Bounds

Finding orientations of supersingular elliptic curves and quaternion orders

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