scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

RAMANUJAN INVARIANTS FOR DISCRIMINANTS CONGRUENT TO 5 (mod 24)

2012 ◽  
Vol 08 (01) ◽  
pp. 265-287 ◽  
Author(s):  
ELISAVET KONSTANTINOU ◽  
ARISTIDES KONTOGEORGIS
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Modular Functions ◽  
Minimal Polynomials ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Lecture Notes ◽  
Yield Class ◽  
Class Fields

In this paper we compute the minimal polynomials of Ramanujan values [Formula: see text] for discriminants D ≡ 5 ( mod 24). Our method is based on Shimura Reciprocity Law as which was made computationally explicit by Gee and Stevenhagen in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)]. However, since these Ramanujan values are not class invariants, we present a modification of the method used in [Generating class fields using Shimura reciprocity, in Algorithmic Number Theory, Lecture Notes in Computer Science, Vol. 1423 (Springer, Berlin, 1998), pp. 441–453; MR MR1726092 (2000m:11112)] which can be applied on modular functions that do not necessarily yield class invariants.

scholarly journals Hyperloops do not threaten the notion of an effective procedure

2017 ◽  
Author(s):  
Tim Button
Keyword(s):  
Computer Science ◽  
Effective Procedure ◽  
Computational Process ◽  
Philosophical Significance ◽  
Lecture Notes ◽  
Effective Computability

This paper develops my (BJPS 2009) criticisms of the philosophical significance of a certain sort of infinitary computational process, a hyperloop. I start by considering whether hyperloops suggest that ‘effectively computable’ is vague (in some sense). I then consider and criticise two arguments by Hogarth, who maintains that hyperloops undermine the very idea of effective computability. I conclude that hyperloops, on their own, cannot threaten the notion of an effective procedure.Published in Lecture Notes in Computer Science 5635: 68–78.

scholarly journals A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2294
Author(s):  
Hari Mohan Srivastava
Keyword(s):  
Number Theory ◽  
Special Functions ◽  
Applied Mathematics ◽  
Analytic Number Theory ◽  
Review Article ◽  
Mathematical Functions ◽  
Analytic Number ◽  
Recent Developments ◽  
Transcendental Functions ◽  
Mathematical Sciences

Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.

scholarly journals Idempotent Factorizations of Square-free Integers

2019 ◽  
Author(s):  
Barry Fagin
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Infinite Number ◽  
Preliminary Results ◽  
Analytical Results ◽  
Hard Problems ◽  
Positive Integers

We explore the class of positive integers n that admit idempotent factorizations n=pq such that lambda(n) divides (p-1)(q-1), where lambda(n) is the Carmichael lambda function. Idempotent factorizations with p and q prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p and/or q. Idempotent factorizations are exactly those p and q that generate correctly functioning keys in the RSA 2-prime protocol with n as the modulus. While the resulting p and q have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution.

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