Number Of Distinct Prime Divisors Research Articles

Explicit estimates of the number of distinct prime divisors of binomial coefficients

UDC 511 We propose explicit estimates of the number of distinct prime divisors of a binomial coefficient through the explicit generalizations of some principal existing results. We also prove interesting number-theoretical propositions for the terms of a sequence of primes.

Some computational results on a conjecture of de Polignac about numbers of the form p + 2k

Let U be the set of positive odd numbers that can not be written in the form p+2k. Recently, by analyzing possible prime divisors of b, Chen proved b≥11184810 and ω(b)≥7 if an arithmetic progression a(modb) is in U, with ω(b)=7 if and only if b=11184810, where ω(n) is the number of distinct prime divisors of n. In this paper, we take a computational approach to prove b≥11184810 and provide all possible values of a if a(mod11184810) is in U. Moreover, we explicitly construct nontrivial arithmetic progressions a(modb) in U with ω(b)=8, 9, 10, or 11, and provide potential nontrivial arithmetic progressions a(modb) in U such that ω(b)=s for any fixed s≥12. Furthermore, we improve the upper bound estimate of numbers of the form p+2k by Habsieger and Roblot in 2006 to 0.490341088858244 by enhancing their algorithm and employing GPU computation.

Fourier coefficients of cusp forms on special sequences

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms [Formula: see text] and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.

On the index of special perfect polynomials

We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $\omega(d) \geq 9$, and $\deg(d) \geq 258$, where $d := \gcd(Q^2,\sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ \sigma(A)=A$ in the polynomial ring ${\mathbb{F}}_2[x]$. The function $\sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174].

Distinguishing newforms by the prime divisors of their Fourier coefficients

Given two non-CM newforms with integral Fourier coefficients, in this paper we study the number of distinct prime divisors of their Fourier coefficients in a probability way. Based on a multivariate version of the Erdős-Kac theorem, using the Galois representations attached to newforms and the effective Chebotarev density theorem, and assuming the generalized Riemann hypothesis, we show that the distribution of the number of distinct primes dividing the Fourier coefficients behaves like the standard multivariate normal distribution if these newforms are not twists of each other. As a consequence, we prove a multiplicity one result for modular forms under the generalized Riemann hypothesis.

On the minimum cut-sets of the power graph of a finite cyclic group

The power graph [Formula: see text] of a finite group [Formula: see text] is the simple graph with vertex set [Formula: see text], in which two distinct vertices are adjacent if one of them is a power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. For [Formula: see text], the minimum cut-sets of [Formula: see text] are characterized in [S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Vertex connectivity of the power graph of a finite cyclic group, Discrete Appl. Math. 266 (2019) 259–271]. In this paper, for [Formula: see text], we identify certain cut-sets of [Formula: see text] such that any minimum cut-set of [Formula: see text] must be one of them.

A localized Erdős-Kac theorem for $\omega_y(p+a)$

Let $\omega_y(n)$ denote the number of distinct prime divisors of $n$ less than $y$. Suppose $y_n$ is an increasing sequence of positive real numbers satisfying $\log y_n = o(\log\log n)$. In this paper, we prove an Erd\"{o}s-Kac theorem for the distribution of $\omega_{y_n}(p+a)$, where $p$ runs over all prime numbers and $a$ is a fixed integer. We also highlight the connection between the distribution of $\omega_y(p-1)$ and Ihara's conjectures on Euler-Kronecker constants.

The maximal size of a minimal generating set

Abstract A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfies $m(G) \leqslant a \cdot \delta (G)^b$ , for $\delta (G) = \sum _{p \text { prime}} m(G_p)$ , where $G_p$ is a Sylow p-subgroup of G. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$ ). In particular, we prove that there exist $a,b> 0$ such that any finite simple group G of Lie type of rank r over the field $\mathbb {F}_{p^f}$ satisfies $r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$ , where $\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over $\mathbb {F}_{p^f}$ has size at most $ar^b + \omega (f)$ .

Asymptotic formula for the multiplicative function $\frac{d(n)}{k^{\omega(n)}}$

For a fixed integer $k$, we define the multiplicative function \[D_{k,\omega}(n) := \frac{d(n)}{k^{\omega(n)}}, \]where $d(n)$ is the divisor function and $\omega (n)$ is the number of distinct prime divisors of $n$. The main purpose of this paper is the study of the mean value of the function $D_{k,\omega}(n)$ by using elementary methods.

On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms

Abstract In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 \epsilon>0 , there exist infinitely many natural numbers 𝑛 such that τ ⁢ ( p n ) \tau(p^{n}) has at least 2 ( 1 - ϵ ) ⁢ log ⁡ n log ⁡ log ⁡ n 2^{(1-\epsilon)\frac{\log n}{\log\log n}} distinct prime factors for almost all primes 𝑝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above.

Tamagawa number divisibility of centralL-values of twists of the Fermat elliptic curve

Given any integer N>1 prime to 3, we denote by C N the elliptic curve x 3 +y 3 =N. We first study the 3-adic valuation of the algebraic part of the value of the Hasse–Weil L-function L(C N ,s) of C N over ℚ at s=1, and we exhibit a relation between the 3-part of its Tate–Shafarevich group and the number of distinct prime divisors of N which are inert in the imaginary quadratic field K=ℚ(-3). In the case where L(C N ,1)≠0 and N is a product of split primes in K, we show that the order of the Tate–Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.

A localized Erdős-Kac theorem

Let ω_y (n) be the number of distinct prime divisors of n not exceeding y. If y_n is an increasing function of n such that log y_n = o(log n), we study the distribution of ω_{y_n} (n) and establish an analog of the Erdős-Kac theorem for this function. En route, we also prove a variant central limit theorem for random variables, which are not necessarily independent, but are well approximated by independent random variables.

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

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]

Ranks in the family of hyperelliptic Jacobians of y2 = x5 + ax

Consider the hyperelliptic curves Ca:y2=x5+ax defined over Q, and their Jacobians Ja. Without loss of generality a is a non-zero 8th power free integer. Our aim is to obtain upper bounds for ra:=rankJa(Q). In particular, we would like to find infinite subfamily of Ja with rank 0. We show that under certain assumptions on the quartic field Q(−a4), ra≤ω(2a)+1 (if a>0), and ra≤ω(2a)+2 (if a<0) where ω(m) is the number of distinct prime divisors of m. We also generalize this result to the ranks of the Jacobians of y2=x(xn+a). Moreover, we prove that if p≡3(mod8) is a prime then rp,r−p≤1, r−2p≤2, and if p≡11,19(mod32) then rp=0, and consequently Cp(Q)={∞,(0,0)} for such primes. We also make numerical computations of ranks and rational points, and put a few conjectures.

New Proof That the Sum of the Reciprocals of Primes Diverges

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.

On the minimum degree of the power graph of a finite cyclic group

The power graph [Formula: see text] of a finite group [Formula: see text] is the undirected simple graph whose vertex set is [Formula: see text], in which two distinct vertices are adjacent if one of them is an integral power of the other. For an integer [Formula: see text], let [Formula: see text] denote the cyclic group of order [Formula: see text] and let [Formula: see text] be the number of distinct prime divisors of [Formula: see text]. The minimum degree [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [R. P. Panda and K. V. Krishna, On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups, Comm. Algebra 46(7) (2018) 3182–3197]. For [Formula: see text], under certain conditions involving the prime divisors of [Formula: see text], we identify at most [Formula: see text] vertices such that [Formula: see text] is equal to the degree of at least one of these vertices. If [Formula: see text], or that [Formula: see text] is a product of distinct primes, we are able to identify two such vertices without any condition on the prime divisors of [Formula: see text].

Combinatorial identities and Titchmarsh’s divisor problem for multiplicative functions

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and $h\in\mathbb{Z}\setminus\{0\}$. We consider in particular the special cases where $f$ is the generalized divisor function $\tau_z$ with $z\in\mathbb{C}$, and the characteristic function of sums of two squares (or more generally, ideal norms of abelian extensions). As another application, we deduce a full asymptotic expansion in the generalized Titchmarsh divisor problem $\sum_{|h|<n\leq x,\,\omega(n)=k} \tau(n-h)$, where $\omega(n)$ counts the number of distinct prime divisors of $n$, thus extending a result of Fouvry and Bombieri-Friedlander-Iwaniec. We present two different proofs: The first relies on an effective combinatorial formula of Heath-Brown's type for the divisor function $\tau_\alpha$ with $\alpha\in\mathbb{Q}$, and an interpolation argument in the $z$-variable for weighted mean values of $\tau_z$. The second is based on an identity of Linnik type for $\tau_z$ and the well-factorability of friable numbers.

VALUE PATTERNS OF MULTIPLICATIVE FUNCTIONS AND RELATED SEQUENCES

We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set$A$whose indicator function is ‘approximately multiplicative’ and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern$n+1\in A$,$n+2\in A$,$n+3\in A$is positive as long as$A$has density greater than$\frac{1}{3}$. Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of$A$having density exactly$\frac{1}{3}$, below which one would need nontrivial information on the local distribution of$A$in Bohr sets to proceed. We apply our results first to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors$P^{+}(n)$,$P^{+}(n+1),P^{+}(n+2)$of three consecutive integers. Second, we show that the tuple$(\unicode[STIX]{x1D714}(n+1),\unicode[STIX]{x1D714}(n+2),\unicode[STIX]{x1D714}(n+3))~(\text{mod}~3)$takes all the$27$possible patterns in$(\mathbb{Z}/3\mathbb{Z})^{3}$with positive lower density, with$\unicode[STIX]{x1D714}(n)$being the number of distinct prime divisors. We also prove a theorem concerning longer patterns$n+i\in A_{i}$,$i=1,\ldots ,k$in approximately multiplicative sets$A_{i}$having large enough densities, generalizing some results of Hildebrand on his ‘stable sets conjecture’. Finally, we consider the sign patterns of the Liouville function$\unicode[STIX]{x1D706}$and show that there are at least$24$patterns of length$5$that occur with positive upper density. In all the proofs, we make extensive use of recent ideas concerning correlations of multiplicative functions.

Residue races of the number of prime divisors function

We investigate the distribution of the function ω(n), the number of distinct prime divisors of n, in residue classes modulo q for natural numbers q greater than 2. In particular we ask ‘prime number races’ style questions, as suggested by Coons and Dahmen in their paper ‘On the residue class distribution of the number of prime divisors of an integer’.

A remark on the number of distinct prime divisors of integers

We study the asymptotic formula for the sum $\sum_{n\leqslant x}\o(n)$ where $\o(n)$ denotes the number of distinct prime divisors of $n$, and we perform some computations which detect curve patterns in the distribution of a related sequence.