Number Of Distinct Prime Divisors Research Articles

Conjugacy Class Lengths of Finite Groups with Prime Graph a Tree

For a finite group G, we write to denote the prime divisor set of the various conjugacy class lengths of G and the maximum number of distinct prime divisors of a single conjugacy class length of G. It is a famous open problem that can be bounded by . Let G be an almost simple group G such that the graph built on element orders is a tree. By using Lucido’s classification theorem, we prove except possibly when G is isomorphic to , where p is an odd prime and α is a field automorphism of odd prime order f. In the exceptional case, . Combining with our known result, we also prove that for a finite group G with a forest, the inequality is true.

Conjugacy class lengths of finite groups with disconnected prime graph

For a nite group G, we write c (G) to denote the prime divisor set of the various conjugacy class lengths of G and c (G) the maximum number of distinct prime divisors of a single conjugacy class length ofG. It is conjectured that the inequalityjc (G)j 3c (G) is true for every nite group. In this note, we prove that the inequality jc (G)j 2c (G) is true when the ( element order ) prime graph ( G) is disconnected by using Gruenberg-Kegel classication theorem and Casolo’s result. Mathematics Subject Classication: 20Dxx (Primary), 20E15 (Secondary)

On non-abelian Schur groups

A finite group G is called Schur, if every Schur ring over G is associated in a natural way with a regular subgroup of sym (G) that is isomorphic to G. We prove that any non-abelian Schur group G is metabelian and the number of distinct prime divisors of the order of G does not exceed 7.

On additive combinatorics of permutations of ℤ<sub>n</sub>

Special issue PRIMA 2013 Let ℤ<sub>n</sub> denote the ring of integers modulo n. A permutation of ℤ<sub>n</sub> is a sequence of n distinct elements of ℤ<sub>n</sub>. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤ<sub>n</sub>, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that no sum of two distinct permutations in the collection is a permutation. Let the sizes be denoted by s(n) and t(n) respectively. The case when n is even is trivial in both the cases, with s(n)=1 and t(n)=n!. For n odd, we prove (nφ(n))/2<sup>k</sup>≤s(n)≤n!· 2<sup>-(n-1)/2</sup>((n-1)/2)! and 2<sup>(n-1)/2</sup>·(n-1 / 2)!≤t(n)≤ 2<sup>k</sup>·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.

ON THE GENERAL QUADRATIC GAUSS SUMS WEIGHTED BY CHARACTER SUMS OVER A SHORT INTERVAL

Abstract. By using the analytic methods, the mean value of the generalquadratic Gauss sums weighted by the first power mean of character sumsover a short interval is investigated. Several sharp asymptotic formulaeare obtained, which show that these sums enjoy good distributive prop-erties. Moreover, interesting connections among them are established. 1. Introduction and main resultsFor any integer n, the general quadratic Gauss sums G(n,χ;q) is defined asG(n,χ;q) =X qa=1 χ(a)ena 2 q,where e(y) = e 2πiy . This summation is very important, since it is the general-ization of the classical quadratic Gauss sums. But we still know little aboutthe properties of G(n,χ;q), we do not even know how large G(n,χ;q) is. Sincethe value of |G(n,χ;q)| is irregular as χ varies, one can only get some upperbound estimates. For example, for any integer n with (n,q) = 1, from thegeneral result of Cochrane and Zheng [1] we can deduce that|G(n,χ;q)| ≤ 2 ω(q) q 12 ,where ω(q) denotes the number of distinct prime divisors of q. The case thatq is prime is due to A. Weil [2].However, weighted sums [4] involving G(n,χ;q) enjoys many good valuedistribution properties, through which interesting connections among them areestablished. Now we shall use analytic methods to study the mean value of thegeneral quadratic Gauss sums weighted by the first power mean of character

Note on the [formula omitted]th power means of distinct prime divisors of composite positive integers

Suppose that h is a positive integer. For an integer n ≥ 2 , define P h ( n ) = ( 1 ω ( n ) ∑ p ∣ n p prime p h ) 1 / h , where ω ( n ) denotes the number of distinct prime divisors of n . Let A h ( x ) be the set of all positive integers n ≤ x with ω ( n ) > 1 such that P h ( n ) is prime and P h ( n ) ∣ n . In this paper, we prove that x exp ( 2 h log x log log x ) ≤ | A h ( x ) | ≤ x exp ( ( 1 / 2 ) log x log log x ) , which generalizes a result of Luca and Pappalardi for h = 1 .

ON VALUES OF d(n!)/m!, ϕ(n!)/m! AND σ(n!)/m!

For f one of the classical arithmetic functions d, ϕ and σ, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.

GAUSSIAN LAWS ON DRINFELD MODULES

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

On the Energy of Unitary Cayley Graphs

In this note we obtain the energy of unitary Cayley graph $X_{n}$ which extends a result of R. Balakrishnan for power of a prime and also determine when they are hyperenergetic. We also prove that ${E(X_{n})\over 2(n-1)}\geq{2^{k}\over 4k}$, where $k$ is the number of distinct prime divisors of $n$. Thus the ratio ${E(X_{n})\over 2(n-1)}$, measuring the degree of hyperenergeticity of $X_{n}$, grows exponentially with $k$.

A new upper bound for the cross number of finite abelian groups

In this paper, building among others on earlier works by U. Krause and C. Zahlten (dealing with the case of cyclic groups), we obtain a new upper bound for the little cross number valid in the general case of arbitrary finite abelian groups. Given a finite abelian group, this upper bound appears to depend only on the rank and the number of distinct prime divisors of the exponent. The main theorem of this paper allows us, among other consequences, to prove that a classical conjecture concerning the cross and little cross numbers of finite abelian groups holds asymptotically in at least two different directions.

A Carlitz module analogue of a conjecture of Erdos and Pomerance

Let A = F q [ T ] A=\mathbb {F}_q[T] be the ring of polynomials over the finite field F q \mathbb {F}_q and 0 ≠ a ∈ A 0 \neq a \in A . Let C C be the A A -Carlitz module. For a monic polynomial m ∈ A m\in A , let C ( A / m A ) C(A/mA) and a ¯ \bar {a} be the reductions of C C and a a modulo m A mA respectively. Let f a ( m ) f_a(m) be the monic generator of the ideal { f ∈ A , C f ( a ¯ ) = 0 ¯ } \{f\in A, C_f(\bar {a}) =\bar {0}\} on C ( A / m A ) C(A/mA) . We denote by ω ( f a ( m ) ) \omega (f_a(m)) the number of distinct monic irreducible factors of f a ( m ) f_a(m) . If q ≠ 2 q\neq 2 or q = 2 q=2 and a ≠ 1 , T a\neq 1, T , or ( 1 + T ) (1+T) , we prove that there exists a normal distribution for the quantity \[ ω ( f a ( m ) ) − 1 2 ( log ⁡ deg ⁡ m ) 2 1 3 ( log ⁡ deg ⁡ m ) 3 / 2 . \frac {\omega (f_a(m))-\frac {1}{2}(\log \deg m)^2}{\frac {1}{\sqrt {3}}{(\log \deg m)^{3/2}}}. \] This result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of b b modulo n n , where b b is an integer with | b | > 1 |b|>1 , and n n a positive integer.

Prime Divisors Of Some Recurrence Sequence

In this paper, we study various arithmetic properties of the sequence (an)n≥1 satisfying the recurrence relation an = nan–1 + 1, n = 2, 3,..., with the initial term a1 = 0. In particular, we estimate the number of solutions of various congruences with this sequence and the number of distinct prime divisors of its first N terms.

Perfect polynomials over $\mathbb F_4$ with less than five prime factors

A perfect polynomial A\in\mathbb{F}_4[x] is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials A\in\mathbb{F}_4[x] with exactly 3 prime divisors, i.e., of the form A=P^aQ^bR^c where P,Q,R\in\mathbb{F}_4[x] are irreducible and a,b,c are positive integers. We characterize the perfect polynomials A with 4 prime divisors such that one of them has degree 1 . Assume that A has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.

The greatest prime divisor of a product of terms in an arithmetic progression

For an integer v > 1, we denote by co(v) and P(v) the number of distinct prime divisors of v and the greatest prime factor of v, respectively, and we put co(l) = O, P(1) = 1. Further we write zra(v) for the number of primes ~ k. Let d = 1. A well known theorem of Sylvester [15] states that

Arithmetic properties of the Ramanujan function

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and $$P(\tau )(p)\tau (p^2 )\tau (p^3 )) > (1 + o(1))\frac{{\log \log p\log \log \log p}}{{\log \log \log \log p}}$$ for every primep with τ(ρ) ≠ 0.

A prime analogue of the Erdös--Pomerance conjecture for elliptic curves

Let E/{\mathbb Q} be an elliptic curve of rank \ge 1 and b\in E({\mathbb Q}) a rational point of infinite order. For a prime p of good reduction, let g_b(p) be the order of the cyclic group generated by the reduction \bar b of b modulo p . We denote by \omega(g_b(p)) the number of distinct prime divisors of g_b(p) . Assuming the GRH, we show that the normal order of \omega(g_b(p)) is \log \log p . We also prove conditionally that there exists a normal distribution for the quantity \frac{\omega(g_b(p)) - \log \log p}{\sqrt{\log \log p}}. The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of \omega(f_a(n)) , where a is a natural number > 1 and f_a(n) the order of a modulo n .

Prime analogues of the Erdős–Kac theorem for elliptic curves

Let E / Q be an elliptic curve. For a prime p of good reduction, let E ( F p ) be the set of rational points defined over the finite field F p . We denote by ω ( # E ( F p ) ) , the number of distinct prime divisors of # E ( F p ) . We prove that the quantity (assuming the GRH if E is non-CM) ω ( # E ( F p ) ) − log log p log log p distributes normally. This result can be viewed as a “prime analogue” of the Erdős–Kac theorem. We also study the normal distribution of the number of distinct prime factors of the exponent of E ( F p ) .

A weighted Turán sieve method

We develop a weighted Turán sieve method and applied it to study the number of distinct prime divisors of f ( p ) where p is a prime and f ( x ) a polynomial with integer coefficients.

On Fibonacci numbers with few prime divisors

If $n$ is a positive integer, write $F_n$ for the $n$th Fibonacci number, and $\omega(n)$ for the number of distinct prime divisors of $n$. We give a description of Fibonacci numbers satisfying $\omega(F_n) \leq 2$. Moreover, we prove that the inequality $\omega(F_n) \geq (\log n)^{\log 2 + o(1)}$ holds for almost all $n$. We conjecture that $\omega(F_n) \gg \log n$ for composite $n$, and give a heuristic argument in support of this conjecture.

Number of prime divisors in a product of terms of an arithmetic progression

Let d ≥ 1, k ≥ 3, n ≥ 1 be integers with gcd(n, d) = 1. We denote ∆ = ∆(n, d, k) = n(n+ d) · · · (n+ (k − 1)d). For an integer ν > 1, we write ω(ν) and P (ν) for the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively. Further we put ω(1) = 0 and P (1) = 1. For l coprime to d, we write π(ν, d, l) for the number of primes ≤ ν and congruent to l modulo d. Further, we denote by πd(ν) for the number of primes ≤ ν and coprime to d. The letter p always denote a prime number. Let W (∆) denote the number of terms in ∆ divisible by a prime > k. We observe that every prime exceeding k divides at most one term of ∆. Therefore we have W (∆) ≤ ω(∆)− πd(k). (1) If max(n, d) ≤ k, we see that n+(k−1)d ≤ k and therefore no term of ∆ is divisible by more than one prime exceeding k. Thus W (∆) = ω(∆)− πd(k) if max(n, d) ≤ k. (2)