scholarly journals Beyond Knuth's Notation for Unimaginable Numbers within Computational Number Theory

2022 ◽  
pp. 55-73
Author(s):  
Antonino LEONARDIS ◽  
Gianfranco D'ATRI ◽  
Fabio CALDAROLA
Keyword(s):  
Number Theory ◽  
Computational Number Theory

scholarly journals On a New Formula for Fibonacci’s Family m-step Numbers and Some Applications

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 805 ◽  
Author(s):  
Monther Rashed Alfuraidan ◽  
Ibrahim Nabeel Joudah
Keyword(s):  
Number Theory ◽  
Time Complexity ◽  
Matrix Multiplication ◽  
Square Roots ◽  
Computational Number Theory ◽  
The Matrix ◽  
Theory Application ◽  
Constant Coefficients ◽  
New Formula

In this work, we obtain a new formula for Fibonacci’s family m-step sequences. We use our formula to find the nth term with less time complexity than the matrix multiplication method. Then, we extend our results for all linear homogeneous recurrence m-step relations with constant coefficients by using the last few terms of its corresponding Fibonacci’s family m-step sequence. As a computational number theory application, we develop a method to estimate the square roots.

NEW INVARIANTS OF SIMPLE KNOTS

2008 ◽  
Vol 17 (03) ◽  
pp. 337-350
Author(s):  
C. KEARTON ◽  
S. M. J. WILSON
Keyword(s):  
Number Theory ◽  
Computational Number Theory

Our longterm plan is to classify knot modules and pairings by utilizing the power of computational number theory. The first step in this is to define invariants for which any given value arises from only finitely many modules: this is the purpose of the present paper.

scholarly journals Computing Modular Polynomials

2005 ◽  
Vol 8 ◽  
pp. 195-204 ◽  
Author(s):  
Denis Charles ◽  
Kristin Lauter
Keyword(s):  
Number Theory ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Probabilistic Algorithm ◽  
Computational Number Theory ◽  
Modular Polynomials

AbstractThis paper presents a new probabilistic algorithm to compute modular polynomials modulo a prime. Modular polynomials parameterize pairs of isogenous elliptic curves, and are useful in many aspects of computational number theory and cryptography. The algorithm presented here has the distinguishing feature that it does not involve the computation of Fourier coefficients of modular forms. The need to compute the exponentially large integral coefficients is avoided by working directly modulo a prime, and computing isogenies between elliptic curves via Vélu's formulas.

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