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Root Sets of Polynomials Modulo Prime Powers

2001
*
Journal of combinatorial theory. Series A
*

INTRODUCTION A subset R

doi:10.1006/jcta.2000.3069
fatcat:spoe2t5klnbzhn3mo6bvrz2cgu
*of*ZÂnZ is a*root**set**modulo*n if there is a*polynomial*over Z whose*roots**modulo*n are exactly the elements*of*R. ... /Z be a p-*root**set**modulo*p k . Then the*set*S=[s 0 , s 1 , ...] is a*root**set**modulo*p k&+(T, k) . Proposition 3 . 3 Let S=[a 0 , a 1 , ...]/Z be a*root**set**modulo*p k . ...##
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Frobenius and his Density theorem for primes

2003
*
Resonance
*

We saw one who made sense

doi:10.1007/bf02839049
fatcat:5hxkg6uhcncobeeaa6yi53lj6q
*of**prime*numbers being dense. This was the great George Frobenius! ... The*set**of*odd*primes**modulo*which the*polynomial*X~+I has*roots*,consists precisely*of*all*primes*in the arithmetic progression 4n+1. ... The*set**of*odd*primes**modulo*which the*polynomial*X 2 + 1 has*roots*, consists precisely*of*all*primes*in the arithmetic progression 4n + 1. ...##
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Efficient Optimization Method for Polynomial Selection

다항식 선택을 위한 효율적인 최적화 기법

2016
*
Journal of the Korea Institute of Information Security and Cryptology
*

다항식 선택을 위한 효율적인 최적화 기법

However, optimization

doi:10.13089/jkiisc.2016.26.3.631
fatcat:lamb2kqciffi7p5lf363fa6bee
*of*selected*polynomial*in CADO-NFS is an immense procedure which takes 90%*of*time in total*polynomial*selection. ...*Polynomial*selection in CADO-NFS can be divided into two stages -*polynomial*selection, and optimization*of*selected*polynomial*. ... many*roots**modulo**prime**powers*for first smallest*primes*. ...##
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Rotations and Translations of Number Field Sieve Polynomials
[chapter]

2003
*
Lecture Notes in Computer Science
*

We present an algorithm that finds

doi:10.1007/978-3-540-40061-5_18
fatcat:oudjkqluxnhl3ghsw7dukgq6pa
*polynomials*with many*roots**modulo*many*primes*by rotating candidate Number Field Sieve*polynomials*using the Chinese Remainder Theorem. ... We also present an algorithm that finds a*polynomial*with small coefficients among all integral translations*of*X*of*a given*polynomial*in Z Z [X]. ... (Rotation) Let f ∈ Z Z[X] be a*polynomial**of*degree d, with*root*m*modulo*N . Let S be a finite*set**of**powers**of*distinct*primes*S = {p e1 1 , . . . , p es s } and 0 ≤ r < d. ...##
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Roots of Polynomials Modulo Prime Powers

1997
*
European journal of combinatorics (Print)
*

The theorems

doi:10.1006/eujc.1996.0124
fatcat:w3xx4rojunbdpl6etli2tvvuvy
*of*the next section provide tools that permit the ef ficient computation*of*the number*of**root**sets**modulo*a*prime**power*. ...*Of*course , for a*prime*p , every subset*of*Z p is a*root**set**modulo*p , but , in general , it appears that the property*of*being a*root**set**modulo*n is rare . ... It follows that we may reduce a j*modulo*p k Ϫ e j without changing the*root**set**modulo*p k*of*the*polynomial*. . ...##
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Root optimization of polynomials in the number field sieve

2015
*
Mathematics of Computation
*

It consists

doi:10.1090/s0025-5718-2015-02926-3
fatcat:6idk3q46ung7rf4ojusniryuge
*of*several stages, the first one being*polynomial*selection. The quality*of*the chosen*polynomials*in*polynomial*selection can be modelled in terms*of*size and*root*properties. ... In this paper, we describe some algorithms for selecting*polynomials*with very good*root*properties. ... Conclusion*Root*optimization aims to produce*polynomials*that have many*roots**modulo*small*primes*and*prime**powers*. ...##
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Products of quadratic polynomials with roots modulo any integer

2013
*
International Mathematical Forum
*

We classify products

doi:10.12988/imf.2013.35112
fatcat:y6kz6kmxwfgx7dyx3f34cvxj3y
*of*three quadratic*polynomials*, each irreducible over Q, which are solvable*modulo*m for every integer m > 1 but have no*roots*over the rational numbers. ...*Polynomials*with this property are known as intersective*polynomials*. We use Hensel's Lemma and a refined version*of*Hensel's Lemma to complete the proof. Mathematics Subject Classification: 11R09 ... We employ a refined version*of*Hensel's Lemma in the case*of*a singular*root*, enabling us to lift our solutions*modulo*arbitrarily high*prime**powers*. ...##
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A Congruence Property of Solvable Polynomials
[article]

2022
*
arXiv
*
pre-print

We describe a congruence property

arXiv:2107.01270v4
fatcat:kxbgcmzluvgw5ix6myxotdejhe
*of*solvable*polynomials*over Q, based on the irreducibility*of*cyclotomic*polynomials*over number fields that meet certain conditions. ... Acknowledgments by Author: I am very grateful to Keith Conrad for his helpful comments on earlier drafts*of*this paper. ... Nonetheless, the theorem gives us a property*of*solvable*polynomials*that depends only on the discriminant*of*the*polynomial*and its factorization*modulo*rational*primes*, rather than on the*roots**of*the ...##
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Positive lower density for prime divisors of generic linear recurrences
[article]

2021
*
arXiv
*
pre-print

We prove, under the generalized Riemann hypothesis, that the lower density

arXiv:2102.04042v1
fatcat:oxhm3skwgrgytmtsjss2wmitlu
*of*the*set**of**primes*which divide at least one element*of*the sequence (a_n) is positive. ... Let d ≥ 3 be an integer and let P ∈ℤ[x] be a*polynomial**of*degree d whose Galois group is S_d. Let (a_n) be a linearly recuresive sequence*of*integers which has P as its characteristic*polynomial*. ... Let S denote the*set**of**primes*p such that P factorizes as the product*of**polynomials**of*degree 1 and d − 1*modulo*p. ...##
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Squarefree values of trinomial discriminants

2015
*
LMS Journal of Computation and Mathematics
*

The

doi:10.1112/s1461157014000436
fatcat:qxwayfuotnaipn3c3rf7c6m4bm
*set**of**primes*whose squares can divide these sporadic values as$n$varies seems to be independent*of*$m$, and this*set*can be seen as a generalization*of*the Wieferich*primes*, those*primes*$p$such that ... We provide heuristics for the density*of*these sporadic*primes*and the density*of*squarefree values*of*these trinomial discriminants. ... The genesis*of*this work took place at the 1999 and 2000 Western Number Theory Conferences in Asilomar and San Diego, respectively; we thank the organizers*of*those conferences and their problem sessions ...##
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Cyclotomy and cyclotomic polynomials

1999
*
Resonance
*

Sum

doi:10.1007/bf02838673
fatcat:z5z2lrgx5reb3ikionrpmir7pm
*of*Primitive*Roots*For a*prime*number p, Gauss defined a primitive*root**modulo*p to be an integer a whose order*modulo*p is p -1. ... Hence the sum*of*all the primitive*roots**modulo*p is simply the sum*of*the*roots**of*Op-1*modulo*p. ... by the*set**of**polynomials*over a finite field! ...##
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Linear Divisibility Sequences

1937
*
Transactions of the American Mathematical Society
*

*polynomial*whose

*roots*are the sth

*powers*

*of*the

*roots*offx), and p a

*prime*number. ... Let fw(x) = (x-«i") • • • ix-ak") be the

*polynomial*whose

*roots*are the 5th

*powers*

*of*the

*roots*

*of*f(x), and let Dw be its discriminant. DM/D is clearly an integer. ...

##
###
Linear divisibility sequences

1937
*
Transactions of the American Mathematical Society
*

*polynomial*whose

*roots*are the sth

*powers*

*of*the

*roots*offx), and p a

*prime*number. ... Let fw(x) = (x-«i") • • • ix-ak") be the

*polynomial*whose

*roots*are the 5th

*powers*

*of*the

*roots*

*of*f(x), and let Dw be its discriminant. DM/D is clearly an integer. ...

##
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Die Ganzen Zahlen hat Gott gemacht

2001
*
Resonance
*

Regarding

doi:10.1007/bf02837737
fatcat:gutmlc7wpzei7mzcpl75kuftxy
*roots**of*a*polynomial**modulo*a*prime*, there is following general result due to Lagrange: Lemma 2.1. Let p be a*prime*number and let P(X) E Z[X] be*of*degree n. ... Note that we have observed earlier that any non-constant integral*polynomial*has a*root**modulo*infinitely many*primes*. ...##
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An Indexing for Quadratic Residues Modulo N and a Non-uniform Efficient Decoding Algorithm
[article]

2018
*
arXiv
*
pre-print

We present an indexing for the

arXiv:1805.04731v2
fatcat:smironradzbhnjid777w62xfmq
*set**of*quadratic residues*modulo*N that is decodable in*polynomial*time on the size*of*N, given the factorization*of*N. ... An indexing*of*a finite*set*S is a bijection D : {1,...,|S|}→ S. ... On the other hand, if the factor is a*power**of*an odd*prime*number p k i i , we need to know a square*root*y i ∈ Z * p k i i*of*z*modulo*p k i i . ...
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