scholarly journals Korselt rational bases of prime powers

2019 ◽  
Vol 56 (4) ◽  
pp. 388-403
Author(s):  
Nejib Ghanmi
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Prime Divisor

Abstract Let N be a positive integer, be a subset of ℚ and . N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α2p − α1 divides α2N − α1 for every prime divisor p of N. By the Korselt set of N over , we mean the set of all such that N is an α-Korselt number. In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of is studied where we prove that is empty if and only if l = 2. Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

scholarly journals On the integers represented by x4 − y4

2007 ◽  
Vol 76 (1) ◽  
pp. 133-136 ◽  
Author(s):  
Andrzej Dąbrowski
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Diophantine Equation ◽  
Trivial Solution ◽  
Effective Constant

Let p be a prime number ≥ 5, and n a positive integer > 1. This note is concerned with the diophantine equation x4 − y4 = nzp. We prove that, under certain conditions on n, this equation has no non-trivial solution in Z if p ≥ C(n), where C(n) is an effective constant.

scholarly journals On a density problem of Erdös

1999 ◽  
Vol 22 (3) ◽  
pp. 655-658 ◽  
Author(s):  
Safwan Akbik
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Density Problem

For a positive integern, letP(n)denotes the largest prime divisor ofnand define the set:𝒮(x)=𝒮={n≤x:n   does not divide   P(n)!}. Paul Erdös has proposed that|S|=o(x)asx→∞, where|S|is the number ofn∈S. This was proved by Ilias Kastanas. In this paper we will show the stronger result that|S|=O(xe−1/4logx).

THE KORSELT SET OF pq

2012 ◽  
Vol 08 (02) ◽  
pp. 299-309 ◽  
Author(s):  
OTHMAN ECHI ◽  
NEJIB GHANMI
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Prime Numbers ◽  
Composite Number

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and N - α is a multiple of p - α for each prime divisor p of N. By the Korselt set of N, we mean the set of all α ∈ ℤ\{0} such that N is a Kα-number; this set will be denoted by [Formula: see text]. Given a squarefree composite number, it is not easy to provide its Korselt set and Korselt weight both theoretically and computationally. The simplest kind of squarefree composite number is the product of two distinct prime numbers. Even for this kind of numbers, the Korselt set is far from being characterized. Let p, q be two distinct prime numbers. This paper sheds some light on [Formula: see text].

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

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.

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

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 certain modular determinants

1940 ◽  
Vol 32 ◽  
pp. xxiii-xxx
Author(s):  
H. W. Turnbull
Keyword(s):  
Positive Integer ◽  
Prime Number

An interesting determinant occurs in the fifth volume of Muir's History1. It iswhere n = ½ (p – 1), and ars is the smallest positive integer such thatrars = s (mod p), (1) p being any odd prime number. It is evident that each element ars is unique and non zero. For p = 5, 7, 11 the determinants are (2) respectively, and their values are– 5 , 72, 114.

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