Results 31 to 40 of about 53,772 (244)

Security Estimates for Quadratic Field Based Cryptosystems [PDF]

, 2010
We describe implementations for solving the discrete logarithm problem in the class group of an imaginary quadratic field and in the infrastructure of a real quadratic field.
Alan, Silverster K., Biasse, Jean-François, Michael, Jacobson John   +2 more
core   +4 more sources

A p-adic lower bound for a linear form in logarithms

International Journal of Number Theory, 2022
Linear forms in logarithms have an important role in the theory of Diophantine equations. In this paper, we prove explicit [Formula: see text]-adic lower bounds for linear forms in [Formula: see text]-adic logarithms of rational numbers using Padé approximations of the second kind.
Seppälä Louna, Palojärvi Neea
openaire   +4 more sources

On sums of k-generalized Fibonacci and k-generalized Lucas numbers as first and second kinds of Thabit numbers [PDF]

Notes on Number Theory and Discrete Mathematics
Let (Fᵣ⁽ᵏ⁾)ᵣ≥2-k and (Lᵣ⁽ᵏ⁾)ᵣ≥2-k be generalizations of the Fibonacci and Lucas sequences, where k≥2. For these sequences the initial k terms are 0,0,...,0, 1 and 0,0,...,2,1, and each subsequent term is the sum of the preceding k terms.
Hunar Sherzad Taher, Saroj Kumar Dash
doaj   +1 more source

Linear forms in logarithms and exponential Diophantine equations [PDF]

Hardy-Ramanujan Journal, 2020
This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.
openaire   +3 more sources

Tubular approaches to Baker's method for curves and varieties

, 2019
Baker's method, relying on estimates on linear forms in logarithms of algebraic numbers, allows one to prove in several situations the effective finiteness of integral points on varieties.
Fourn, Samuel Le
core   +2 more sources

Polynomial-Time Amoeba Neighborhood Membership and Faster Localized Solving [PDF]

, 2013
We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number
A. Dickenstein, A.G. Khovanskii, A.J. Sommese, C. Beltrán, C. D’Andrea, CTD Ng, D.A. Plaisted, D.J. Bates, F. Bihan, G. Mikhalkin, G.M. Ziegler, I.M. Gel’fand, I.Z. Emiris, J. Verschelde, J.-C. Faugère, J.A. Loera De, L. Bettale, L. Khachiyan, L. Nilsson, L.J. Huan, M. Grötschel, M. Nisse, M. Shub, M. Shub, M. Shub, M. Shub, M.R. Garey, P. Bürgisser, P. Gritzmann, P. Gritzmann, P. Gritzmann, R.M. Karp, S. Arora, Steve Smale, W. Hao, Y. Ning   +35 more
core   +1 more source

On soft singularities at three loops and beyond [PDF]

, 2009
We report on further progress in understanding soft singularities of massless gauge theory scattering amplitudes. Recently, a set of equations was derived based on Sudakov factorization, constraining the soft anomalous dimension matrix of multi-leg ...
A Armoni, A Banfi, A Bassetto, A Ferroglia, A Ferroglia, A Mitov, A Sen, A Sen, AB Goncharov, AH Mueller, AM Polyakov, AV Kotikov, AV Kotikov, C Bogner, DA Kosower, DA Kosower, E Gardi, E Laenen, E Remiddi, Einan Gardi, G Sterman, G Sterman, G Sterman, GP Korchemsky, GP Korchemsky, GP Korchemsky, GP Korchemsky, H Contopanagos, IA Korchemskaya, J Botts, J Frenkel, JA Azcárraga de, JAM Vermaseren, JC Collins, JC Collins, JC Collins, JGM Gatheral, L Magnea, L Magnea, L Magnea, Lance J. Dixon, LF Alday, LJ Dixon, LJ Dixon, Lorenzo Magnea, M Beneke, M Czakon, N Kidonakis, N Kidonakis, N Kidonakis, N Kidonakis, RA Brandt, RA Brandt, S Catani, S Catani, S Catani, S Moch, SD Badger, SM Aybat, SM Aybat, SV Ivanov, T Becher, T Becher, T Becher, T Gehrmann, TO Eynck, VA Smirnov, VS Dotsenko, YL Dokshitzer, Z Bern, Z Bern, Z Bern, Z Bern, Z Bern   +73 more
core   +3 more sources

Repdigits as Euler functions of Lucas numbers

Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2016
We prove some results about the structure of all Lucas numbers whose Euler function is a repdigit in base 10. For example, we show that if Ln is such a Lucas number, then n < 10111 is of the form p or p2, where p3 | 10p-1 -1.
Bravo Jhon J., Faye Bernadette, Luca Florian, Tall Amadou   +3 more
doaj   +1 more source

S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points

, 2017
Let $S$ be a finite set of primes. The $S$-part $[m]_S$ of a non-zero integer $m$ is the largest positive divisor of $m$ that is composed of primes from $S$.
Bugeaud, Yann, Evertse, Jan-Hendrik, Győry, Kálmán   +2 more
core   +1 more source

On the Diophantine Equation $\left(9d^2 + 1\right)^x + \left(16d^2 - 1\right)^y = (5d)^z$ Regarding Terai's Conjecture

Journal of New Theory
This study proves that the Diophantine equation $\left(9d^2+1\right)^x+\left(16d^2-1\right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$.
Murat Alan, Tuba Çokoksen
doaj   +1 more source