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Let [Formula: see text] be a fixed positive number. Define [Formula: see text] as the number of positive integers [Formula: see text] having no prime factors [Formula: see text], and define [Formula: see text] as the number of positive integers [Formula: see text] having [Formula: see text] prime factors [Formula: see text], with all the other prime factors [Formula: see text]. In this paper, we give an asymptotic estimate for the ratio [Formula: see text], provided that [Formula: see text], [Formula: see text], and [Formula: see text] as [Formula: see text]. Also, combining this estimate with conventional ones for [Formula: see text], we provide sharp estimates for [Formula: see text].

We examine two counting problems that seem very group-theoretic on the surface but, on closer examination, turn out to concern integers with restrictions on their prime factors. First, given an odd prime $q$ and a finite abelian $q$-group $H$, we consider the set of integers $n\le x$ such that the Sylow $q$-subgroup of the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is isomorphic to $H$. We show that the counting function of this set of integers is asymptotic to $K x(\log\log x)^\ell/(\log x)^{1/(q-1)}$ for explicit constants $K$ and $\ell$ depending on $q$ and $H$. Second, we consider the set of integers $n\le x$ such that the multiplicative group $(\mathbb Z/n\mathbb Z)^\times$ is "maximally non-cyclic", that is, such that all of its prime-power subgroups are elementary groups. We show that the counting function of this set of integers is asymptotic to $A x/(\log x)^{1-\xi}$ for an explicit constant $A$, where $\xi$ is Artin's constant. As it turns out, both of these group-theoretic problems can be reduced to problems of counting integers with restrictions on their prime factors, allowing them to be addressed by classical techniques of analytic number theory., Comment: 21 pages

We examine oscillations in a number of sums of arithmetic functions involving [Formula: see text], the total number of prime factors of [Formula: see text], and [Formula: see text], the number of distinct prime factors of [Formula: see text]. In particular, we examine oscillations in [Formula: see text] and in [Formula: see text] for [Formula: see text], and in [Formula: see text]. We show for example that each of the inequalities [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] is true infinitely often, disproving some hypotheses of Sun.

We prove that for the Steinhaus Random Variable [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denotes the number of prime factors of [Formula: see text].

Denote by [Formula: see text] the set of all primes and by [Formula: see text] the largest prime factor of integer [Formula: see text] with the convention [Formula: see text]. Let [Formula: see text] be the unique positive solution of the equation [Formula: see text] in [Formula: see text]. Very recently Wu proved that for [Formula: see text] there is a constant [Formula: see text] such that for each fixed non-zero integer [Formula: see text] the set [Formula: see text] has relative density 1 in [Formula: see text]. In this paper, we shall further extend the domain of [Formula: see text] at the cost of obtaining a lower bound in place of an asymptotic formula, by showing that for each [Formula: see text] the set [Formula: see text] has relative positive density in [Formula: see text].

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with [Formula: see text] in place of [Formula: see text].

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that, for [Formula: see text] and for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes, which constitutes an extension upon that of Lü.

We consider for integers [Formula: see text] the [Formula: see text]-generalized Fibonacci sequences [Formula: see text], whose first [Formula: see text] terms are [Formula: see text] and each term afterwards is the sum of the preceding [Formula: see text] terms. We give a lower bound for the largest prime factor of the sum of two terms in [Formula: see text]. As a consequence of our main result, for every fixed finite set of primes [Formula: see text], there are only finitely many positive integers [Formula: see text] and [Formula: see text]-integers which are a non-trivial sum of two [Formula: see text]-Fibonacci numbers, and all these are effectively computable.

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight.

For each integer [Formula: see text], let [Formula: see text] be the index of composition of [Formula: see text], where [Formula: see text]. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain an [Formula: see text]-result for the error term of the sum [Formula: see text]

A Carmichael number is a composite number [Formula: see text] such that [Formula: see text] divides [Formula: see text] for all integers [Formula: see text] coprime to [Formula: see text]. Korselt discovered that [Formula: see text] is a Carmichael number if and only if [Formula: see text] is square-free and [Formula: see text] for each prime divisor [Formula: see text] of [Formula: see text]. Let [Formula: see text], a [Formula: see text]-number is defined to be a composite number [Formula: see text], such that [Formula: see text] and [Formula: see text] for each prime [Formula: see text]. The set of all [Formula: see text] such that [Formula: see text] is a [Formula: see text]-number is called the Korselt set of [Formula: see text] and we denote this set by [Formula: see text]. In this paper, we investigate some properties of [Formula: see text] when [Formula: see text] is a power of a prime.

In 2014, Wang and Cai established the following harmonic congruence for any odd prime [Formula: see text] and positive integer [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] denote the set of positive integers which are prime to [Formula: see text]. In this paper, we obtain an unexpected congruence for distinct odd primes [Formula: see text], [Formula: see text] and positive integers [Formula: see text], [Formula: see text] and the necessary and sufficient condition for [Formula: see text] Finally, we raise a conjecture that for [Formula: see text] and odd prime power [Formula: see text], [Formula: see text], [Formula: see text] However, we fail to prove it even for [Formula: see text] with three distinct prime factors.

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

Let [Formula: see text] be an additively written finite cyclic group of order [Formula: see text] and let [Formula: see text] be a minimal zero-sum sequence with elements of [Formula: see text], i.e. the sum of elements of [Formula: see text] is zero, but no proper nontrivial subsequence of [Formula: see text] has sum zero. [Formula: see text] is called unsplittable if there do not exist an element [Formula: see text] in [Formula: see text] and two elements [Formula: see text] in [Formula: see text] such that [Formula: see text] and the new sequence [Formula: see text] is still a minimal zero-sum sequence. In this paper, we investigate long unsplittable minimal zero-sum sequences over [Formula: see text]. Our main result characterizes the structures of all such sequences [Formula: see text] and shows that the index of [Formula: see text] is at most 2, provided that the length of [Formula: see text] is greater than or equal to [Formula: see text] where [Formula: see text] is a positive integer with least prime divisor greater than [Formula: see text].

In this paper, we prove that, for any positive constants [Formula: see text] and [Formula: see text] and every large enough [Formula: see text], the interval [Formula: see text] contains numbers whose all prime factors are smaller than [Formula: see text].

In this paper, we prove a conjecture of Sun concerning the arithmetic progression of integers which contains no integers of the form [Formula: see text]. In fact, we show that [Formula: see text] has at least two distinct prime factors whenever [Formula: see text], where [Formula: see text] is a Fibonacci number and [Formula: see text] and [Formula: see text] are suitable positive integers.

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to Oϵ(n1 + ϵ) for arbitrary ϵ > 0 assuming the GRH.

Let $p$ be a fixed prime, and let $v(a)$ stand for the exponent of $p$ in the prime factorization of the integer $a$. Let $f$ and $g$ be two monic polynomials with integer coefficients and nonzero resultant $r$. Write $S$ for the maximum of $v(\gcd (f(n), g(n)))$ over all integers $n$. It is known that $S \le v(r)$. We give various lower and upper bounds for the least possible value of $v(r)-S$ provided that a given power $p^s$ divides both $f(n)$ and $g(n)$ for all $n$. In particular, the least possible value is $ps^2-s$ for $s\le p$ and is asymptotically $(p-1)s^2$ for large $s$., Comment: 12 pages

We consider the problem of finding small prime gaps in various sets [Formula: see text]. Following the work of Goldston–Pintz–Yıldırım, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting qn denote the nth prime in [Formula: see text], we will establish that for any small constant ϵ > 0, the set {qn | qn+1 - qn ≤ ϵ log n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao, we will also demonstrate that [Formula: see text] has bounded prime gaps. Specific examples, such as the case where [Formula: see text] is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.

Let α ∈ ℤ\{0}. A composite number N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and p - α divides N - α for each prime divisor p of N. The set of all α ∈ ℤ\{0} such that N is a Kα-number is called the Korselt set of N and is denoted by 𝒦𝒮(N). In this paper, we study the Korselt set of q2, where q is prime. We describe in detail how to obtain 𝒦𝒮(q2), compute the cardinality of 𝒦𝒮(q2), and answer some questions related to 𝒦𝒮(q2).

We provide irreducibility conditions for polynomials of the form f(X) + pkg(X), with f and g relatively prime polynomials with integer coefficients, deg f < deg g, p a prime number and k a positive integer. In particular, we prove that if k is prime to deg g - deg f and pk exceeds a certain bound depending on the coefficients of f and g, then f(X) + pkg(X) is irreducible over ℚ.

We study the asymptotic behavior of the sequence with general term consisting of the ratio An by Gn, the arithmetic and geometric means of the prime numbers p1, p2, …, pn, respectively, in which, pn denotes the nth prime number.

For each integer n ≥ 2, let [Formula: see text] be the index of composition of n, where γ(n) ≔ ∏p∣np. The index of composition of an integer measures the multiplicity of its prime factors. In this paper, we obtain a new asymptotic formula of the sum ∑n≤xλ-k(n). Furthermore, we improve the error term under the Riemann Hypothesis.

A number is cubefull if the exponent of every prime in its prime factorization is at least three. In this paper, we give, for a fixed l, the number of pairs of cubefull numbers n, n + l such that n is less than a given quantity.

We prove Liouville's type identities on sums of even integer functions ranging over sets involving relatively prime pairs.

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].

We examine the problem of determining the solutions to the Diophantine equation \[ n(n+d)\cdots (n+(k-1)d) = by^{2}, \] where d is fixed and the largest prime divisor of b is no more than Ck. Here, C is fixed but arbitrary. Under some rather minor conditions, it is shown that there are finitely many solutions that can be effectively computed. Some new, largely combinatorial, ideas are introduced into the general theory to handle the case of arbitrary C as considered here.

In this paper, using the theory of double coverings of cyclotomic fields, we give a formula for [Formula: see text], where K = ℚ(ζn), G = Gal (K/ℚ), 𝔽2 = ℤ/2ℤ and UK is the unit group of K. We explicitly determine all the cyclotomic fields K = ℚ(ζn) such that [Formula: see text]. Then we apply it to the unit square problem raised in [Y. Li and X. Zhang, Global unit squares and local unit squares, J. Number Theory128 (2008) 2687–2694]. In particular, we prove that the unit square problem does not hold for ℚ(ζn) if n has more than three distinct prime factors, i.e. for each odd prime p, there exists a unit, which is a square in all local fields ℚ(ζn)v with v | p but not a square in ℚ(ζn), if n has more than three distinct prime factors.

We prove that the Diophantine equation $p_1^{r_1} \cdots p_k^{r_k} = \left( \begin{array}{cc}r_1 + \cdots + r_k\\ r_1, \ldots, r_k \end{array} \right)$ has only finitely many positive integer solutions k, p1, …, pk, r1, …, rk, where p1, …, pk are distinct primes. If a positive integer n has prime factorization $p_1^{r_1}\cdots p_k^{r_k}$, then $ \left(\begin{array}{cc}{r_1} \+ \cdots \+{r_k}} \\ {{r_1}, \ldots \, {r_k}} \end{array} \right)$ represents the number of ordered factorizations of n into prime parts. Hence, solutions to the above Diophantine equation are designated as prime-perfect numbers.

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [nc] such that p + 2 = Pr, where r is the least positive integer satisfying certain inequalities. In particular for [Formula: see text] we have r = 5. This result constitutes an improvement upon that of T. P. Peneva.

Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcm θ∈Âi(θ), where i(θ) = gcd x∈ℤFθ(x) and Fθ(x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρK(p) defined as the number of [Formula: see text] such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρK(p) are stated.

Let N denote a sufficiently large integer satisfying N ≡ 4 (mod 24), and Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the equation [Formula: see text] is solvable in one prime and three P42, or in four P13. These results constitute improvements upon that of Heath-Brown and Tolev.

We prove that, for any ∊ > 0, there exists a constant C > 0 such that the interval [Formula: see text] contains numbers whose all prime factors are smaller than [Formula: see text].

A prime p is elite if all sufficiently large Fermat numbers Fn = 22n + 1 are quadratic nonresidues modulo p. In contrast, p is anti-elite if all sufficiently large Fn are quadratic residues modulo p. The sequence Fn modulo p is necessarily periodic. We give a sequence of pairwise coprime integers whose prime factors are each elite or anti-elite with period four.

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and p - α divides N - α for each prime divisor p of N. We are concerned, here, with both a numerical and theoretical study of composite squarefree Korselt numbers. The paper contains two main results. The first one shows that for α ∈ ℤ\{0}, the following properties hold: (i) If α ≤ 1, then each composite squarefree Kα-number has at least three prime factors. (ii) Suppose that α > 1. Let p < q be two prime numbers and N ≔ pq. If N is an α-Korselt number, then p < q ≤ 4α - 3. In particular, there are only finitely many α-Korselt numbers with exactly two prime factors. Let α ∈ ℕ\{0}; by an α-Williams number (Wα-number, for short) we mean a positive integer which is both a Kα-number and a K-α-number. Our second main result shows that if p, 3p - 2, 3p + 2 are all prime, then their product is a (3p)-Williams number.

We study the solubility of the binary additive equation $[m^c] + [p^c] = n$, where $m$ is an integer, $p$ is a prime number, and $c$ is a fixed real number in the range $1 < c < 3/2$., 10 pages

In this paper, we show that the number of real quadratic fields 𝕂 of discriminant Δ𝕂 < x whose class group has an element of order g (with g even) is ≥ x1/g/5 if x > x0, uniformly for positive integers g ≤ ( log log x)/(8 log log log x). We also apply the result to find real quadratic number fields whose class numbers have many prime factors.

We consider the notion of a balanced set modulo N. A nonempty set S of residues modulo N is balanced if for each x ∈ S, there is a d with 0 < d ≤ N/2 such that x ± d mod N both lie in S. We define α(N) to be the minimum cardinality of a balanced set modulo N. This notion arises in the context of a two-player game that we introduce and has interesting connections to the prime factorization of N. We demonstrate that for p prime, α(p) = Θ( log p), giving an explicit algorithmic upper bound and a lower bound using finite field theory and show that for N composite, α(N) = min p|Nα(p).

We obtain new upper bounds for the mean squared modulus of sums ∑n∈ℕAnχ(n), where the sequence (An) is fixed and the variable χ belongs to the set of non-principal Dirichlet characters for some modulus q. It is assumed that, for some M, some complex sequence (cm) satisfying cm = 0 for m ∉ (M/2,M], and some α(x) and β(y) (smooth functions with compact support), one has An = ∑uvm = n α(u)β(v)cm (n ∈ ℕ). There is a natural analogy between the bounds obtained and bounds on mean values of Dirichlet polynomials previously obtained by Deshouillers and Iwaniec. Our proofs make use of results from the spectral theory of automorphic functions, including the bound of Kim and Sarnak for the eigenvalues of Hecke operators acting on certain spaces of Maass cusp forms. The results depend on the size of P, the largest prime factor of q, and improve as log q(P) is diminished. In separate work, Harman has given an application of our results to the theory of Carmichael numbers.

We show that the set of prime factors of almost all integers are "Poisson distributed", and that this remains true (appropriately formulated) even when we restrict the number of prime factors of the integer. Our results have inspired analogous results about the distribution of cycle lengths of permutations.

We show that for any ∊ > 0, there exists c > 0, such that for all x sufficiently large, there are x1/2 ( log x)- log 4 - o(1) integers [Formula: see text], all of whose prime factors are [Formula: see text].

Kevin Ford has pointed out that the proof of Theorem 3 of [1] contains several significant mistakes (perhaps I did not check it over carefully enough because it is the most easily proved result in the paper!). We give here a correct proof of a slightly modified version of that theorem. Almost all integers up to x have ∼ log log x distinct prime factors. In our article [1], we examined the distribution of the sizes of those prime factors, and Theorem 3 looked at the integers with far fewer prime factors than is typical. If Sk(x) denotes the set of integers n ≤ x which have exactly k distinct prime factors then it is well known that