scholarly journals Modified Cyclotomic Polynomial and Its Irreducibility

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 343
Author(s):  
Ki-Suk Lee ◽  
Sung-Mo Yang ◽  
Soon-Mo Jung
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Irreducible Polynomials

Finding irreducible polynomials over Q (or over Z ) is not always easy. However, it is well-known that the mth cyclotomic polynomials are irreducible over Q . In this paper, we define the mth modified cyclotomic polynomials and we get more irreducible polynomials over Q systematically by using the modified cyclotomic polynomials. Since not all modified cyclotomic polynomials are irreducible, a criterion to decide the irreducibility of those polynomials is studied. Also, we count the number of irreducible mth modified cyclotomic polynomials when m = p α with p a prime number and α a positive integer.

Remarks on the flatness of ternary cyclotomic polynomials

2017 ◽  
Vol 13 (02) ◽  
pp. 529-547 ◽  
Author(s):  
Bin Zhang
Keyword(s):  
Positive Integer ◽  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Polynomial Formula

For a positive integer [Formula: see text], the [Formula: see text]th cyclotomic polynomial [Formula: see text] is called flat if its coefficients are [Formula: see text], [Formula: see text]. Let [Formula: see text] be odd primes with [Formula: see text] for some positive integer [Formula: see text]. In this paper, we classify all the flat cyclotomic polynomials [Formula: see text] when [Formula: see text], [Formula: see text] and [Formula: see text].

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 The order of magnitude of the mth coefficients of cyclotomic polynomials

1985 ◽  
Vol 27 ◽  
pp. 143-159 ◽  
Author(s):  
H. L. Montgomery ◽  
R. C. Vaughan
Keyword(s):  
Cyclotomic Polynomial ◽  
Cyclotomic Polynomials ◽  
Order Of Magnitude ◽  
Euler’S Function ◽  
Image Position

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler's function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

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 The distribution of irreducible polynomials in several indeterminates over a finite field

1968 ◽  
Vol 16 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Stephen D. Cohen
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Equivalence Class ◽  
Ordered Set ◽  
Irreducible Polynomials ◽  
Totally Positive

We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.

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.

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 $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$.

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