scholarly journals Prime divisors of some shifted products

2005 ◽  
Vol 2005 (19) ◽  
pp. 3057-3073
Author(s):  
Eric Levieil ◽  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Prime Divisors ◽  
Positive Integers

We study prime divisors of various sequences of positive integersA(n)+1,n=1,…,N, such that the ratiosa(n)=A(n)/A(n−1)have some number-theoretic or combinatorial meaning. In the casea(n)=n, we obviously haveA(n)=n!, for which several new results about prime divisors ofn!+1have recently been obtained.

NOTE ON LEHMER–PIERCE SEQUENCES WITH THE SAME PRIME DIVISORS

2017 ◽  
Vol 97 (1) ◽  
pp. 11-14
Author(s):  
M. SKAŁBA
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Prime Divisors ◽  
Positive Integers

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$. Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$, then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$.

scholarly journals On a relation between the Fitting length of a soluble group and the number of conjugacy classes of its maximal nilpotent subgroups

1969 ◽  
Vol 1 (1) ◽  
pp. 3-10 ◽  
Author(s):  
H. Lausch ◽  
A. Makan
Keyword(s):  
Soluble Group ◽  
Conjugacy Classes ◽  
Finite Soluble Group ◽  
Prime Divisors ◽  
Soluble Groups ◽  
Odd Order ◽  
Fitting Length ◽  
Qualitative Nature ◽  
Positive Integers

In a finite soluble group G, the Fitting (or nilpotency) length h(G) can be considered as a measure for how strongly G deviates from being nilpotent. As another measure for this, the number v(G) of conjugacy classes of the maximal nilpotent subgroups of G may be taken. It is shown that there exists an integer-valued function f on the set of positive integers such that h(G) ≦ f(v(G)) for all finite (soluble) groups of odd order. Moreover, if all prime divisors of the order of G are greater than v(G)(v(G) - l)/2, then h(G) ≦3. The bound f(v(G)) is just of qualitative nature and by far not best possible. For v(G) = 2, h(G) = 3, some statements are made about the structure of G.

scholarly journals Extensions of some formulae of A. Selberg

1985 ◽  
Vol 8 (2) ◽  
pp. 283-302 ◽  
Author(s):  
Claudia A. Spiro
Keyword(s):  
Arithmetic Progression ◽  
Arbitrary Number ◽  
Error Term ◽  
Fixed Number ◽  
Prime Divisors ◽  
Positive Integers ◽  
Number Of Prime Divisors

This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

A note on weakly prime-additive numbers

2021 ◽  
pp. 1-4
Author(s):  
Jin-Hui Fang
Keyword(s):  
Positive Integer ◽  
Prime Divisors ◽  
Prime Factors ◽  
Positive Integers

A positive integer [Formula: see text] is called weakly prime-additive if [Formula: see text] has at least two distinct prime divisors and there exist distinct prime divisors [Formula: see text] of [Formula: see text] and positive integers [Formula: see text] such that [Formula: see text]. It is easy to see that [Formula: see text]. In this paper, intrigued by De Koninck and Luca’s work, we further determine all weakly prime-additive numbers [Formula: see text] such that [Formula: see text], where [Formula: see text] are distinct odd prime factors of [Formula: see text].

Definability and decidability issues in extensions of the integers with the divisibility predicate

1996 ◽  
Vol 61 (2) ◽  
pp. 515-540 ◽  
Author(s):  
Patrick Cegielski ◽  
Yuri Matiyasevich ◽  
Denis Richard
Keyword(s):  
Computational Complexity ◽  
Alternative Proof ◽  
Order Structure ◽  
Prime Divisors ◽  
Additive Theory ◽  
First Order ◽  
Decision Method ◽  
Definable Relations ◽  
Positive Integers ◽  
Number Of Prime Divisors

AbstractLet be a first-order structure; we denote by DEF() the set of all first-order definable relations and functions within . Let π be any one-to-one function from ℕ into the set of prime integers.Let ∣ and • be respectively the divisibility relation and multiplication as function. We show that the sets DEF(ℕ, π, ∣) and DEF(ℕ, π, •) are equal. However there exists function π such that the set DEF(ℕ, +, ∣), or, equivalently, DEF(ℕ, π, •) is not equal to DEF(ℕ, +, •). Nevertheless, in all cases there is an {π, •}-definable and hence also {π, |}-definable structure over π which is isomorphic to 〈ℕ, +, •〉. Hence theories TH(ℕ, π, ∣) and TH(ℕ, π, •) are undecidable.The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice.The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables.

On the solvability of the simultaneous Pell equations x2 − ay2 = 1 and y2 − bz2 = v12

2021 ◽  
pp. 1-12
Author(s):  
Ruiqin Fu ◽  
Hai Yang
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Prime Divisors ◽  
Positive Integer Solution ◽  
Pell Equations ◽  
Integer Solutions ◽  
Least Solution ◽  
Simultaneous Pell Equations ◽  
Necessary And Sufficient ◽  
Positive Integers

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

scholarly journals PRIME DIVISORS OF LUCAS SEQUENCES AND A CONJECTURE OF SKAŁBA

2005 ◽  
Vol 01 (04) ◽  
pp. 583-591 ◽  
Author(s):  
FLORIAN LUCA ◽  
PANTELIMON STĂNICĂ
Keyword(s):  
Prime Divisors ◽  
Lucas Sequences ◽  
Lucas Sequence ◽  
Almost All ◽  
Positive Integers

In this paper, we give some heuristics suggesting that if (un)n≥0 is the Lucas sequence given by un = (an - 1)/(a - 1), where a > 1 is an integer, then ω(un) ≥ (1 + o(1)) log n log log n holds for almost all positive integers n.

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