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 New Proof That the Sum of the Reciprocals of Primes Diverges

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1414
Author(s):  
Vicente Jara-Vera ◽  
Carmen Sánchez-Ávila
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Lattice Points ◽  
Elementary Approach ◽  
Prime Divisors

In this paper, we give a new proof of the divergence of the sum of the reciprocals of primes using the number of distinct prime divisors of positive integer n, and the placement of lattice points on a hyperbola given by n=pr with prime number p. We also offer both a new expression of the average sum of the number of distinct prime divisors, and a new proof of its divergence, which is very intriguing by its elementary approach.

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

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].

scholarly journals On simultaneous rational approximation to a p-adic number and its integral powers

2011 ◽  
Vol 54 (3) ◽  
pp. 599-612 ◽  
Author(s):  
Yann Bugeaud ◽  
Natalia Budarina ◽  
Detta Dickinson ◽  
Hugh O'Donnell
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Rational Approximation ◽  
Real Numbers ◽  
The Real ◽  
Positive Integers ◽  
Simultaneous Rational Approximation

AbstractLet p be a prime number. For a positive integer n and a p-adic number ξ, let λn(ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ‖qξ‖p,‖qξ2‖p,…,‖qξn‖p are all less than q−λ−1. Here, ‖x‖p denotes the infimum of |x−n|p as n runs through the integers. We study the set of values taken by the function λn.

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 On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 288
Author(s):  
Atsushi Yamagami ◽  
Kazuki Taniguchi
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Prime Number ◽  
Special Issue ◽  
Pascal’S Triangle ◽  
Mersenne Number ◽  
Modulo P ◽  
Pascal's Triangle ◽  
Positive Integers ◽  
Symmetric Property

The Pascal’s triangle is generalized to “the k-Pascal’s triangle” with any integer k ≥ 2 . Let p be any prime number. In this article, we prove that for any positive integers n and e, the n-th row in the p e -Pascal’s triangle consists of integers which are congruent to 1 modulo p if and only if n is of the form p e m − 1 p e − 1 with some integer m ≥ 1 . This is a generalization of a Lucas’ result asserting that the n-th row in the (2-)Pascal’s triangle consists of odd integers if and only if n is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence ( x + 1 ) p e ≡ ( x p + 1 ) p e − 1 ( mod p e ) of binomial expansions which we could prove for any prime number p and any positive integer e. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.

scholarly journals On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1813
Author(s):  
S. Subburam ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Exponential Diophantine Equation ◽  
Research Article ◽  
All Solutions ◽  
Sum Of Products ◽  
Consecutive Integers ◽  
Sums Of Products ◽  
Positive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

scholarly journals A new characterization of L2(p2)

2020 ◽  
Vol 18 (1) ◽  
pp. 907-915
Author(s):  
Zhongbi Wang ◽  
Chao Qin ◽  
Heng Lv ◽  
Yanxiong Yan ◽  
Guiyun Chen
Keyword(s):  
Positive Integer ◽  
Irreducible Character ◽  
Character Degrees ◽  
Prime Divisors

Abstract For a positive integer n and a prime p, let {n}_{p} denote the p-part of n. Let G be a group, \text{cd}(G) the set of all irreducible character degrees of G , \rho (G) the set of all prime divisors of integers in \text{cd}(G) , V(G)=\left\{{p}^{{e}_{p}(G)}|p\in \rho (G)\right\} , where {p}^{{e}_{p}(G)}=\hspace{.25em}\max \hspace{.25em}\{\chi {(1)}_{p}|\chi \in \text{Irr}(G)\}. In this article, it is proved that G\cong {L}_{2}({p}^{2}) if and only if |G|=|{L}_{2}({p}^{2})| and V(G)=V({L}_{2}({p}^{2})) .

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

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