Root Modulo Research Articles

On the existence of certain Lehmer numbers modulo a prime

A Lehmer number modulo an odd prime numberp is a residue class a∈Fp× whose multiplicative inverse ā has opposite parity. Lehmer numbers that are also primitive roots are called Lehmer primitive roots. Analogously, in this article we say that a residue class a∈Fp× is a Lehmer non-primitive root modulop if a is Lehmer number modulo p which is not a primitive root. We provide explicit estimates for the difference between the number of Lehmer non-primitive roots modulo a prime p and their “expected number”, which is p−1−ϕ(p−1)2. Similar explicit bounds are also provided for the number of k-consecutive Lehmer numbers modulo a prime, and k-consecutive Lehmer primitive roots We also prove that for any prime number p>3.05×1014, there exists a Lehmer non-primitive root modulo p. Moreover, we show that for any positive integer k≥2 (respectively, k≥5) and for all primes p≥exp(122k3) (respectively, p≥exp(6.87k)), there exist k consecutive Lehmer numbers modulo p (respectively, k consecutive Lehmer primitive roots modulo p). For large primes p, these theorems generalize two results which were proven in a paper by Cohen and Trudgian appeared in the Journal of Number Theory in 2019.

Ranks of quadratic twists of Jacobians of generalized Mordell curves

Consider a two-parameter family of hyperelliptic curves Cq,b:y2=xq−bq defined over Q, and their Jacobians Jq,b where q is an odd prime and without loss of generality b is a non-zero squarefree integer. The curve Cq,b is a quadratic twist by b of Cq,1 (a generalized Mordell curve of degree q). First, we obtain a few upper bounds for the ranks e.g., if the class number of Q(ζq) is odd, q≡1(mod4) and any prime divisor of 2b not equal to q is a primitive root modulo q then rankJq,b(Q)≤(q−1)/2. Then we focus on q=5 and get the best possible bound (by 1) or even the exact value of rank (0). In particular, we found infinitely many b with any number of prime factors such that rankJ5,b(Q)=0. We deduce as conclusions the complete list (or the bounds for the number) of rational points on C5,b in such cases. Finally, we found for any given q infinitely many non-isomorphic curves Cq,b such that rankJq,b(Q)≥1.

О генерической сложности проблемы извлечения квадратного корня по простому модулю

We study the generic complexity of the problem of finding a square root modulo a prime number. The question about the computational complexity of this problem is still open. However, there are known algorithms (e.g. Cipolla’s algorithm) which are polynomial if the extended Riemann hypothesis holds. We prove that this problem is generically decidable in polynomial time. In fact, this means that Cipolla’s algorithm runs in polynomial time for “almost all” inputs. The notion “almost all” is formalized by introducing the asymptotic density on a set of input data.

Abelian lifts of polynomials

Let p be a prime, and let f∈Z[x] be a non-constant polynomial whose leading coefficient is prime to p and that f has no repeated root modulo p. Suppose the mod-p irreducible factors of f all have the same degree. Then there are infinitely many Abelian extensions of number fields L/K such that OL/pOL≃(Z/p)[x]/f, where OL denotes the ring of integers of L and p⊂OK is a maximal ideal with OK/pOK≃Z/p. Moreover, under the generalized Riemann hypothesis for the Dedekind zeta functions of number fields, for fixed deg⁡(f) we can construct Abelian lifts in polynomial time in log⁡p. This confirms in a stronger form a question raised by Mihăilescu and Vuletescu concerning polynomial cyclic algebras. The proofs make use of a weak form of Artin's conjecture on primitive roots and polynomial time polynomial factoring algorithms over number fields.

On some symmetries of the base n expansion of 1∕m : the class number connection

Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \mathbb{Q}(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Furthermore, we obtain corollaries connecting the $ 3 $ rank of the class number of the real quadratic field $ \mathbb{Q}(\sqrt{m}) $ to the $ 3 $ divisibility of the number of certain quadratic residues modulo $ m $. Secondly, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain distribution results of $ m $ modulo any prime $ p $ that properly divides $ n+1 $. Our result implies that the primes $ m $ for which $ n $ is a primitive root belong to one of two equivalence classes modulo any prime $ p $ as above.

Cyclotomy, cyclotomic cosets and arithmetic properties of some families in 𝔽l[x] 〈xptq−1〉

Arithmetic properties of some families in [Formula: see text] and hence in [Formula: see text] are obtained by using the cyclotomic classes of order 2 with respect to [Formula: see text], where [Formula: see text] is primitive root modulo [Formula: see text], [Formula: see text] and [Formula: see text]. The form of these cyclotomic classes enables us to generalize the results obtained in [S. Jain and S. Batra, Cyclotomy and arithmetic properties of some families in [Formula: see text], Asian-Eur. J. Math. 13(4) (2020) 2050077].

Polynomials over ring of integers of global fields that have roots modulo every finite indexed subgroup

A polynomial with coefficients in the ring of integers OK of a global field K is called intersective if it has a root modulo every finite-indexed subgroup of OK. We prove two criteria for a polynomial f(x)∈OK[x] to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon K and the polynomial f. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.

On polynomials with roots modulo almost all primes

Call a monic integer polynomial <em>exceptional</em> if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials $h$ for which there is an irreducible monic <em>quadra

Explicit upper bound on the least primitive root modulo p2

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

АНАЛИЗ ЛИНЕЙНОЙ СЛОЖНОСТИ Q-ИЧНЫХ ОБОБЩЕННЫХ ЦИКЛОТОМИЧЕСКИХ ПОСЛЕДОВАТЕЛЬНОСТЕЙ ПЕРИОДА pn

The article presents the analysis of the linear complexity of periodic q-ary sequences when changing k of their terms per period. Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. There has been obtained a recurrence relation and an estimate of the change in the linear complexity of these sequences, where q is a primitive root modulo equal to the period of the sequence. It can be inferred from the results that the linear complexity of these sequences does not sign ificantly decrease when k is less than half the period. The study summarizes the results for the binary case obtained earlier.

Relative class numbers inside the $p$th cyclotomic field

For a prime number $p \equiv 3 \;\mbox{mod $4$}$, we write $p=2n\ell^f+1$ for some power $\ell^f$ of an odd prime number $\ell$ and an odd integer $n$ with $\ell \nmid n$. For $0 \leq t \leq f$, let $K_t$ be the imaginary subfield of ${\Bbb Q}(\zeta_p)$ of degree $2\ell^t$ and let $h_t^-$ be the relative class number of $K_t$. We show that for $n=1$ (resp. $n \geq 3$), a prime number $r$ does not divide the ratio $h_t^-/h_{t-1}^-$ when $r$ is a primitive root modulo $\ell^2$ and $r \geq \ell^{f-t}-1$ (resp. $r \geq (n-2)\ell^{f-t}+1$). In particular, for $n=1$ or $3$, the ratio $h_f^-/h_{f-1}^-$ at the top is not divisible by $r$ whenever $r$ is a primitive root modulo $\ell^2$. Further, we show that the $\ell$-part of $h_t^-/h_{t-1}^-$ stabilizes for ``large $t$ under some assumption.

On the distribution of square-full and cube-full primitive roots

A positive integer n is called an r-full integer if for all primes $$p\mid n$$ we have $$p^r\mid n.$$ Let p be an odd prime. For $$\gcd (n,p)=1$$, the smallest positive integer f such that $$n^f\equiv 1\pmod p$$ is called the exponent of n modulo p. If $$f=p-1$$ then n is called a primitive root modulo p. Let $$T_r(n)$$ be the characteristic function of the r-full primitive roots modulo p. In this paper we derive the asymptotic formula for the following sums $$\begin{aligned} \sum _{n\le x}T_2(n),\quad \sum _{n\le x}T_3(n), \end{aligned}$$by using properties of character sums.

PRIMES IN ARITHMETIC PROGRESSIONS AND NONPRIMITIVE ROOTS

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$-near primitive root and we prove a more difficult variant.

Cyclotomic cosets and primitive idempotents in semisimple ring

Let be distinct primes. Let , where at least two of αi’ s are nonzero positive integers. In this article, it is shown that with the help of λ-mapping it is easy to count the number of q-cyclotomic cosets modulo m. The explicit expressions of primitive idempotents are obtained in the semisimple ring Rm over Fq taking q as a primitive root (or a quadratic residue) modulo pi under the condition that , where denote the order of q modulo pi.

Browkin’s discriminator conjecture

Let $q\ge 5$ be a prime and put $q^*=(-1)^{(q-1)/2}\cdot q$. We consider the integer sequence $u_q(1),u_q(2),\ldots,$ with $u_q(j)=(3^j-q^*(-1)^j)/4$. No term in this sequence is repeated and thus for each $n$ there is a smallest integer $m$ such that $u_q(1),\ldots,u_q(n)$ are pairwise incongruent modulo $m$. We write $D_q(n)=m$. The idea of considering the discriminator $D_q(n)$ is due to Browkin (2015) who, in case $3$ is a primitive root modulo $q,$ conjectured that the only values assumed by $D_q(n)$ are powers of $2$ and of $q$. We show that this is true for $n\neq 5$, but false for infinitely many $q$ in case $n=5$. We also determine $D_q(n)$ in case 3 is not a primitive root modulo $q$. Browkin's inspiration for his conjecture came from earlier work of Moree and Zumalacarregui (2016), who determined $D_5(n)$ for $n\ge 1$, thus establishing a conjecture of Salajan. For a fixed prime $q$ their approach is easily generalized, but requires some innovations in order to deal with all primes $q\ge 7$ and all $n\ge 1$. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.

The weight distributions of irreducible cyclic codes of length 2npm

Let [Formula: see text] be a primitive root modulo [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd primes. Let [Formula: see text] be a finite field. For such pair of [Formula: see text] and [Formula: see text], the explicit expressions of minimal and generating polynomials over [Formula: see text] are obtained for all irreducible cyclic codes of length [Formula: see text]. In Sec. 4, it is observed that the weight distributions of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] can be computed easily with the help of the results obtained in [P. Kumar, M. Sangwan and S. K. Arora, The weight distribution of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289]. An explicit formula is also given to compute the weight distributions of irreducible cyclic codes of length [Formula: see text] over [Formula: see text].

On the Class Numbers in the Cyclotomic - and -Extensions of the Field of Rationals

ABSTRACTLet p be a prime number. For each prime number ℓ, we consider the problem whether ℓ divides class numbers of finite subextensions in the cyclotomic -extension of the field of rationals or not. Even in the case where ℓ is a primitive root modulo p2, the problem is not solved in general. In this paper, we focus on the case p = 29 and 31. And we show that ℓ does not divide class numbers of finite subextensions in the cyclotomic - and -extensions of the field of rationals if a prime number ℓ is a primitive root modulo p2.

Square-full primitive roots

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text].

Optimal groups for the $r$-rank Artin Conjecture

For any finitely generated subgroup $\Gamma$ of $\mathbb{Q}^*$, Pappalardi and the first--named author [1] found a formula to compute the density of the primes $\ell$ for which the reduction modulo $\ell$ of $\Gamma$ contains a primitive root modulo $\ell$. They conjectured a characterization of optimal groups, free or torsion, i.e. subgroups with maximal density. In this paper we prove their conjecture and give a similar characterization for optimal positive groups.

Об оценках сложности алгоритмов извлечения квадратных корней в конечных полях и кольцах вычетов

The article presents a review of the known algorithms for extracting square roots of the quadratic residues modulo primes and prime powers. According to the Chinese remainder theorem, such algorithms can be constructed to calculate square roots modulo any integer. These include the Tonelli and Chipolla probabilistic algorithms and deterministic algorithms for calculating the square roots of quadratic residues in residue rings modulo a prime number not congruent to 1 modulo 8, as well as the algorithms for extracting a square root modulo a power of a prime number constructed on their basis. The article presents new algorithms for extracting square roots in finite fields of characteristic congruent to 3 modulo 4, for square roots of residues congruent to 1 modulo 8, and for square roots in residue rings modulo power of two. For probabilistic algorithms, the calculation complexity and the probability of successfully accomplishing the calculation in the first try are estimated. The square roots extraction problem encountered in number-theoretic algorithms of cryptography used in combination with the Chinese remainder theorem for root square extraction in residue rings modulo an arbitrary number is considered. The article presents known probabilistic and deterministic algorithms for extracting square roots modulo a prime number, along with estimating the complexity of the algorithms and the probability of successfully completing the computation in the first try. New implementing in the elliptic curve cryptography algorithms for extracting square roots in finite fields of characteristic congruent to 3 modulo 4 are substantiated, and estimates of their complexity are given. Algorithms for extracting square roots in fields of even order are studied, and estimates of their complexity are given. Algorithms for extracting square roots in residue rings modulo prime powers and modulo power of two are demonstrated.