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### Root Sets of Polynomials Modulo Prime Powers

Davesh Maulik
2001 Journal of combinatorial theory. Series A
INTRODUCTION A subset R 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 .  ...

### Frobenius and his Density theorem for primes

B. Sury
2003 Resonance
We saw one who made sense 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.  ...

### Efficient Optimization Method for Polynomial Selection 다항식 선택을 위한 효율적인 최적화 기법

Suhri Kim, Heetaek Kwon, Yongseong Lee, Nam Su Chang, Kisoon Yoon, Chang Han Kim, Young-Ho Park, Seokhie Hong
2016 Journal of the Korea Institute of Information Security and Cryptology
However, optimization 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.  ...

### Rotations and Translations of Number Field Sieve Polynomials [chapter]

Jason E. Gower
2003 Lecture Notes in Computer Science
We present an algorithm that finds 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.  ...

### Roots of Polynomials Modulo Prime Powers

Bruce Dearden, Jerry Metzger
1997 European journal of combinatorics (Print)
The theorems 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 . .  ...

### Root optimization of polynomials in the number field sieve

Shi Bai, Richard P. Brent, Emmanuel Thomé
2015 Mathematics of Computation
It consists 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.  ...

### Products of quadratic polynomials with roots modulo any integer

Andrea M. Hyde, Blair K. Spearman
2013 International Mathematical Forum
We classify products 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.  ...

### A Congruence Property of Solvable Polynomials [article]

Nicholas Phat Nguyen
2022 arXiv   pre-print
We describe a congruence property 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  ...

### Positive lower density for prime divisors of generic linear recurrences [article]

Olli Järviniemi
2021 arXiv   pre-print
We prove, under the generalized Riemann hypothesis, that the lower density 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.  ...

### Squarefree values of trinomial discriminants

David W. Boyd, Greg Martin, Mark Thom
2015 LMS Journal of Computation and Mathematics
The 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  ...

### Cyclotomy and cyclotomic polynomials

B. Sury
1999 Resonance
Sum 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!  ...

### Linear Divisibility Sequences

Morgan Ward
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

Morgan Ward
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.  ...

### Die Ganzen Zahlen hat Gott gemacht

B. Sury
2001 Resonance
Regarding 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.  ...

### An Indexing for Quadratic Residues Modulo N and a Non-uniform Efficient Decoding Algorithm [article]

Nicollas M. Sdroievski and Murilo V. G. da Silva and André L. Vignatti
2018 arXiv   pre-print
We present an indexing for the 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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