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In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

Let $\mathcal{C}$ be a fusion category over an algebraically closed field $\mathbb{k}$ of arbitrary characteristic. Two numerical invariants of $\mathcal{C}$ , that is, the Casimir number and the determinant of $\mathcal{C}$ are considered in this paper. These two numbers are both positive integers and admit the property that the Grothendieck algebra $(\mathcal{C})\otimes_{\mathbb{Z}}K$ over any field K is semisimple if and only if any of these numbers is not zero in K. This shows that these two numbers have the same prime factors. If moreover $\mathcal{C}$ is pivotal, it gives a numerical criterion that $\mathcal{C}$ is nondegenerate if and only if any of these numbers is not zero in $\mathbb{k}$ . For the case that $\mathcal{C}$ is a spherical fusion category over the field $\mathbb{C}$ of complex numbers, these two numbers and the Frobenius–Schur exponent of $\mathcal{C}$ share the same prime factors. This may be thought of as another version of the Cauchy theorem for spherical fusion categories.

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

We obtain a lower bound on the largest prime factor of the denominator of rational numbers in the Cantor set. This gives a stronger version of a recent result of Schleischitz [‘On intrinsic and extrinsic rational approximation to Cantor sets’, Ergodic Theory Dyn. Syst. to appear] obtained via a different argument.

We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$ . Our estimate is uniform in $x,y,w$ . We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table, which are free of prime factors $\leq w$ , and the number of distinct fractions of the form $(a_{1}a_{2})/(b_{1}b_{2})$ with $1\leq a_{1}\leq b_{1}\leq N$ and $1\leq a_{2}\leq b_{2}\leq N$ .

Recently, Gerhard J. Woeginger [1] gave a survey on the interesting history of results on equiangular n-vertex polygons with edge lengths in arithmetic progression. Such a polygon exists if, and only if, n has at least two distinct prime factors.

Given a positive integer n let ω (n) denote the number of distinct prime factors of n, and let a be fixed positive integer. Extending work of Kubilius, we develop a bivariate probabilistic model to study the joint distribution of the deterministic vectors (ω(n), ω(n + a)), with n ≤ x as x → ∞, where n and n + a belong to a subset of ℕ with suitable properties. We thus establish a quantitative version of a bivariate analogue of the Erdős–Kac theorem on proper subsets of ℕ.We give three applications of this result. First, if y = x0(1) is not too small then we prove (in a quantitative way) that the y-truncated Möbius function μy has small binary autocorrelations. This gives a new proof of a result due to Daboussi and Sarkőzy. Second, if μ(n; u) :=e(uω(n)), where u ∈ ℝ then we show that μ(.; u) also has small binary autocorrelations whenever u = o(1) and $u\sqrt {\mathop {\log }\nolimits_2 x} \to \infty$, as x → ∞. These can be viewed as partial results in the direction of a conjecture of Chowla on binary correlations of the Möbius function.Our final application is related to a problem of Erdős and Mirsky on the number of consecutive integers less than x with the same number of divisors. If $y = x^{{1 \over \beta }}$, where β = β(x) satisfies certain mild growth conditions, we prove a lower bound for the number of consecutive integers n ≤ x that have the same number of y-smooth divisors. Our bound matches the order of magnitude of the one conjectured for the original Erdős-Mirsky problem.

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except for an explicit list of exceptions and that $S$ is always ‘large’ in the sense that $|T|^{1/3}. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that, for every $T$ and every prime divisor $r$ of $|T|$ with $r\neq p$, the order of the Sylow $r$-subgroup of $T$ is at most $|T|^{2\lfloor \log _{r}(4(\ell +1)r)\rfloor /\ell }=|T|^{O(\log _{r}(\ell )/\ell )}$, where $\ell$ is the Lie rank of $T$.

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

A random binary search tree grown from the uniformly random permutation of $[n]$ is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction $c_k$ of vertices of a fixed rank $k\ge 0$ is shown to decay exponentially with $k$. Notoriously hard to compute, the exact fractions $c_k$ had been determined for $k\le 3$ only. We computed $c_4$ and $c_5$ as well; both are ratios of enormous integers, denominator of $c_5$ being $274$ digits long. Prompted by the data, we proved that, in sharp contrast, the largest prime divisor of $c_k$'s denominator is $2^{k+1}+1$ at most. We conjecture that, in fact, the prime divisors of every denominator for $k>1$ form a single interval, from $2$ to the largest prime not exceeding $2^{k+1}+1$., 28 pages

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$, $$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$ This answers a question raised by Cuny and Weber, who showed this result for $L^{p}$, $p>1$.

LetSbe a polynomial ring over a fieldKof characteristic zero and letM⊂Sbe an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case whereMis a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime idealP⊂ScontainingMbe such that the localisationMPhasnon-maximal analytic spread. Our technique describes, in fact, a concrete obstruction forPto be an asymptotic prime divisor ofMwith respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.

In this paper, we prove the decidability of the theory of ℚp in the language (+, −,⋅, 0, 1, Pn(n ∈ ℕ)) expanded by a predicate for the multiplicative subgroup nℤ (where n is a fixed integer). There are two cases: if $v_p \left( n \right) > 0$ then the group determines a cross-section and we get an axiomatization of the theory and a result of quantifier elimination. If $v_p \left( n \right) = 0$, then we use the Mann property of the group to get an axiomatization of the theory.

Let $P(n)$ denote the largest prime factor of an integer $n\geq 2$. In this paper, we study the distribution of the sequence $\{f(P(n)):n\geq 1\}$ over the set of congruence classes modulo an integer $b\geq 2$, where $f$ is a strongly $q$-additive integer-valued function (that is, $f(aq^{j}+b)=f(a)+f(b),$ with $(a,b,j)\in \mathbb{N}^{3}$, $0\leq b). We also show that the sequence $\{{\it\alpha}P(n):n\geq 1,f(P(n))\equiv a\;(\text{mod}~b)\}$ is uniformly distributed modulo 1 if and only if ${\it\alpha}\in \mathbb{R}\!\setminus \!\mathbb{Q}$.

Problems concerning the setof numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identitywhich shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.

In this paper, we prove that, if $N$ is a positive odd number with $r$ distinct prime factors such that $N\mid \sigma (N)$, then $N\lt {2}^{{4}^{r} - {2}^{r} } $ and $N{\mathop{\prod }\nolimits}_{p\mid N} p\lt {2}^{{4}^{r} } $, where $\sigma (N)$ is the sum of all positive divisors of $N$. In particular, these bounds hold if $N$ is an odd perfect number.

Recently, Pollack and Shevelev [‘On perfect and near-perfect numbers’, J. Number Theory 132 (2012), 3037–3046] introduced the concept of near-perfect numbers. A positive integer $n$ is called near-perfect if it is the sum of all but one of its proper divisors. In this paper, we determine all near-perfect numbers with two distinct prime factors.

Given an integer $q\ge 2$, a $q$-normal number is an irrational number $\eta $ such that any preassigned sequence of $\ell $ digits occurs in the $q$-ary expansion of $\eta $ at the expected frequency, namely $1/q^\ell $. In a recent paper we constructed a large family of normal numbers, showing in particular that, if $P(n)$ stands for the largest prime factor of $n$, then the number $0.P(2)P(3)P(4)\ldots ,$ the concatenation of the numbers $P(2), P(3), P(4), \ldots ,$ each represented in base $q$, is a $q$-normal number, thereby answering in the affirmative a question raised by Igor Shparlinski. We also showed that $0.P(2+1)P(3+1)P(5+1) \ldots P(p+1)\ldots ,$ where $p$ runs through the sequence of primes, is a $q$-normal number. Here, we show that, given any fixed integer $k\ge 2$, the numbers $0.P_k(2)P_k(3)P_k(4)\ldots $ and $0. P_k(2+1)P_k(3+1)P_k(5+1) \ldots P_k(p+1)\ldots ,$ where $P_k(n)$ stands for the $k{\rm th}$ largest prime factor of $n$, are $q$-normal numbers. These results are part of more general statements.

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

LetGbe a finited-dimensional classical group andpa prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily ofp-singular elements inG(elements with order divisible byp) that leave invariant a subspaceXof the naturalG-module of dimension greater thand/2 and either act irreducibly onXor preserve a particular decomposition ofXinto two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion inGof these so-calledp-abundantelements is at least an absolute constant multiple of the best currently known lower bound for the proportion of allp-singular elements. From a computational point of view, thep-abundant elements generalise another class ofp-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope thatp-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element isp-abundant, both for a known primepand for the case wherepis not knowna priori.

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed. We obtain a full description in the case in which for each prime divisor p of the order of the group there exists a noncentral conjugacy class of p-power size.

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a homotopy data triple is composite, we show that equivalence of nanophrases under the corresponding homotopy can be calculated just by using the homotopies given by its prime factors., Comment: 33 pages, 6 figures. Second version: Improved abstract and introduction

Given an integer d≥2, a d-normal number, or simply a normal number, is an irrational number whosed-ary expansion is such that any preassigned sequence, of length k≥1, taken within this expansion occurs at the expected limiting frequency, namely 1/dk. Answering questions raised by Igor Shparlinski, we show that 0.P(2)P(3)P(4)…P(n)… and 0.P(2+1)P(3+1)P(5+1)…P(p+1)…, where P(n) stands for the largest prime factor of n, are both normal numbers.

Let G be a finite group, let p be a prime divisor of the order of G and let k(G) be the number of conjugacy classes of G. By disregarding at most finitely many non-solvable p-solvable groups G, we have $k(G)\geq2\smash{\sqrt{p-1}}$ with equality if and only if if $\smash{\sqrt{p-1}}$ is an integer, $G=C_{p}\rtimes\smash{C_{\sqrt{p-1}}}$ and CG(Cp) = Cp. This extends earlier work of Héthelyi, Külshammer, Malle and Keller.

Given a ±1 random completely multiplicative function f, we prove by estimating moments that the limiting distribution of the normalized sum converges to the standard Gaussian distribution as x → ∞ when r restricts summation to n having o(log log log x) prime factors. We also give an upper bound for the large deviations of with the sum restricted to numbers having a fixed number k of prime factors.

This essay demonstrates an application of prime numbers to the development of a calculator program that solves sudoku puzzles. Among the positive integers, the primes—numbers with exactly two divisors, the numbers themselves and 1—are central to our thinking about numbers. They give us a basis for factoring and divisibility and they contribute to the solution of many problems in the mathematical field of number theory. More important for the purposes of this paper, they provide a way of representing numbers uniquely by prime factors. For example, 6221592 = 23 × 32 × 13 × 172 × 23, any other factorisation differing only in the order of factors.

In an earlier paper in the Gazette the authors of define what it means for a matrix in a set M of n × n matrices to be prime, namely if it is not the product of two matrices in M, neither of which is the identity. They then showed that there are exactly two primes in the set M2 of 2 × 2 matrices with non-negative integral entries and unit determinant, namely

Let a,b,c be relatively prime positive integers such that a2+b2=c2 with b even. In 1956 Jeśmanowicz conjectured that the equation ax+by=cz has no solution other than (x,y,z)=(2,2,2) in positive integers. Most of the known results of this conjecture were proved under the assumption that 4 exactly divides b. The main results of this paper include the case where 8 divides b. One of our results treats the case where a has no prime factor congruent to 1 modulo 4, which can be regarded as a relevant analogue of results due to Deng and Cohen concerning the prime factors of b. Furthermore, we examine parities of the three variables x,y,z, and give new triples a,b,c such that the conjecture holds for the case where b is divisible by 8. In particular, to prove our results, we shall show an important result which asserts that if x,y,z are all even, then x/2,y/2,z/2 are all odd. Our methods are based on elementary congruence and several strong results on generalized Fermat equations given by Darmon and Merel.

Using elementary means, we derive an explicit formula for a3(n), the number of 3-core partitions of n, in terms of the prime factorization of 3n+1. Based on this result, we are able to prove several infinite families of arithmetic results involving a3(n), one of which specializes to the recent result of Baruah and Berndt which states that, for all n≥0, a3(4n+1)=a3(n).

Let Γ be an elliptic curve defined over Q, all of whose 2-division points are rational, and let Γb be its quadratic twist by b. Subject to a mild additional condition on Γ, we find the limit of the probability distribution of the dimension of the 2-Selmer group of Γb as the number of prime factors of b increases; and we show that this distribution depends only on whether the 2-Selmer group of Γ has odd or even dimension.

In this note, we show that if we write ⌊en!⌋ = s(n)u(n)2, where s(n) is square-free then has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the mth power-free part of s(n) as n ranges from 1 to N, where m ≥ 3 is a positive integer. As an application of such results, we give an upper bound on the number of n ≤ N such that ⌊en!⌋ is a square.

{3, 4, 5} is perhaps the most famous Pythagorean Triple with interest in such triples dating back many thousands of years to the ancient people of Mesopotamia. In this article, we shall consider such triples, with the restriction that the elements of these triples must not have any common factors they are Primitive Pythagorean Triples (PPTs). In particular, we shall consider the question of how many PPTs a given integer can be a member of. The answer to this simple question is, surprisingly, that a given integer n can play the role of a specified side in either 0 or 2k−1 different PPTs, where k is the number of distinct prime factors of n. Our result is a generalisation of what Fermat grandly called the Fundamental Theorem on right-angled triangles ([2], chapter 5), which states that

A subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in coreG (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.

We restrict primes and prime powers to sets . Let . Then the error in θH(x) has, unconditionally, the expected order of magnitude . However, if then ψH (x) = x log 2 + O (log x). Some reasons for and consequences of these sharp results are explored. A proof is given of the “harmonic prime number theorem”, πH (x)/π (x) → log 2.

In this paper we show that if f (X) ∈; Z [X ] is a nonzero polynomial, then ω(n)/f(n) holds only on a set of n of asymptotic density zero, where for a positive integer n the number ω(n) counts the number of distinct prime factors ofn.

Let $\scr E$ denote the set of irrational numbers whose continued fraction convergents $p_j/q_j$ obey the rule $$\log q_{j+1} \le (\log q_j)^{1 + o(1)} \quad (j \to \infty).$$ Thus $\scr E$ contains the algebraic numbers and its complement has Hausdorff dimension 0. Denoting by $\|\cdots\|$ the distance to the nearest integer, the authors obtain $$\min_{d|n} \|d\theta\|= \frac 1{\tau(n)^{1+o(1)}} \quad \text{a.e.},$$ that is, for a set of natural numbers $n$ of density 1. The proof involves the estimation of exponential sums $$\sum_{n\le x} z^{\Omega(n)}e(n\theta)$$ and an excursion into the theory of modified Dirichlet $L$-functions of the form $$L(s, \chi; y) = \sum_{P^+(n) \le y} \frac{\chi(n)}{n^s},$$ where $P^+(n)$ denotes the greatest prime factor of $n$. For example, the authors prove that $$\sum\Sb \chi \pmod q \\\chi \ne \chi_0\endSb |L(1, \chi; y)|^b \ll \phi(q) (\log\log 2q)^b$$ for any $b \ge 1$. (R.C. Baker)

Let $L$ be an unsplittable, prime, oriented, alternating link type in $S^3$ . Let $D$ be a reduced alternating diagram representing $L$ . We define the Murasugi atoms of $D$ as the oriented link types represented by the prime factors of the Murasugi special components of $D$ . We prove (an invariance theorem) that the collection of Murasugi atoms depends only on $L$ and not on $D$ . This has the following corollary. Let $L$ be as above and assume that $L$ is achiral. Write its HOMFLY polynomial as $P_{L}(v,z)\,{=}\,\sum_{m}^{M} b_{j}(v) z^j$ . Then $b_{M}(v)\,{=}{\pm}\, \beta(v) \beta(v^{-1})$ for some polynomial $\beta(v) \in\mathbb{Z}[v, v^{-1}]$ . As a consequence, the leading coefficient of the Conway polynomial of $L$ is a square (up to sign).

Those who enjoy number patterns will no doubt have had many hours of pleasure exploring the Fibonacci sequence to various moduli, and especially in recognising the regularity with which various prime factors occur in thesequence (see [1]). Gill Hatch’s question is whether the occurrence of prime factors in generalised Fibonacci sequences is similarly predictable. Generalised Fibonacci sequences (Gn), abbreviated to GF sequences, are sequences of positive integers derived from the recurrence relation tn + 2 = tn + 1 + tn. In the case of the Fibonacci sequence (Fn), the first two terms are 1 and 1.

Let p be an odd prime number. By using Iwasawa theory, we shall construct cyclotomic fields whose maximal real subfields have class group with arbitrarily large p-rank and conductor with only four prime factors.

We shall introduce a p-local geometry Δp(G) for the pair (G, p) of a finite group G and a prime divisor p of the order of G, which is constructed by ‘maximal parabolic like’ subgroups of G. Under the natural hypothesis, Δp(G) behaves very much like the building associated with a group of Lie type in characteristic p. We shall also show that, under some hypothesis, Δp(G) is homotopy equivalent to the subgroup complex [Bscr ]cenp(G) which is a more essential part of the p-radical complex [Bscr ]p(G). Some of the p-local sporadic geometries can be understood well in our system.

Let D = p1 … pm, where p1, …, pm are distinct rational primes ≡ 1(mod 8), and m is any positive integer. In this paper, we give a simple combinatorial criterion for the value of the Hecke L-function of the congruent elliptic curve ED2 : y2 = x3 − D2x at s = 1, divided by the period ω defined below, to be exactly divisible by 4m. As a corollary, we obtain a series of non-congruent numbers whose number of prime factors tends to infinity, and for which the corresponding elliptic curves have non-trivial 2-part of Tate–Shafarevich group, which greatly generalizes a result of Razar [8]. Our result is in accord with the predictions of the conjecture of Birch and Swinnerton-Dyer.

Let n = p1p2 … pr be a product of r prime numbers which are not necessarily different. We define then an arithmetic function µm(n) bywhere m is a natural number. We further define the function L(s, µm) by the Dirichlet seriesand will show that L(s, µm), (m ≥ 3), has an infinitely many valued analytic continuation into the half plane Re s > ½.

With one exception, namely {2, 3, 5, 7}, it is impossible to have four consecutive primes p1, p2, p3, p4 with p4 - p1 < 8. An interval of seven or less cannot contain more than three odd numbers unless one of them is a multiple of three. On the other hand, groups of four primes p, p + 2, p + 6, p + 8, usually called prime quadruplets, are fairly common. The first is {5, 7, 11, 13}, followed by {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829} and so on. Just as with prime twins, pairs of primes p, p + 2, it is conjectured that the sequence of prime quadruplets goes on for ever. Indeed, the apparently simpler prime twin conjecture is currently an unsolved problem of mathematics although in 1973, Jing-Run Chen proved a weaker form: There are infinitely many primes p such that p + 2 is either prime or the product of two primes (See Halberstam & Richert [1]). A similar result holds for quadruplets [1, Theorem 10.6]: There exist infinitely many primes p such that (p + 2) (p + 6) (p + 8) has at most 14 prime factors. The prime quadruplet conjecture would then follow if we could reduce ‘14’ to ‘3’ but this seems be a problem of extreme difficulty.

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ε, we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding nε, with the length of the strings tending to infinity with speed log log log log n.

Given a class of combinatorial structures [Cscr ], we consider the quantity N(n, m), the number of multiset constructions [Pscr ] (of [Cscr ]) of size n having exactly m [Cscr ]-components. Under general analytic conditions on the generating function of [Cscr ], we derive precise asymptotic estimates for N(n, m), as n→∞ and m varies through all possible values (in general 1[les ]m[les ]n). In particular, we show that the number of [Cscr ]-components in a random (assuming a uniform probability measure) [Pscr ]-structure of size n obeys asymptotically a convolution law of the Poisson and the geometric distributions. Applications of the results include random mapping patterns, polynomials in finite fields, parameters in additive arithmetical semigroups, etc. This work develops the ‘additive’ counterpart of our previous work on the distribution of the number of prime factors of an integer [20].

The prime factorization of a random integer has a GEM/Poisson-Dirichlet distribution as transparently proved by Donnelly and Grimmett [8]. By similarity to the arc-sine law for the mean distribution of the divisors of a random integer, due to Deshouillers, Dress and Tenenbaum [6] (see also Tenenbaum [24, II.6.2, p. 233]), – the ‘DDT theorem’ – we obtain an arc-sine law in the GEM/Poisson-Dirichlet context. In this context we also investigate the distribution of the number of components larger than ε which correspond to the number of prime factors larger than nε.

We show here that the first order theory of the positive integers equipped with multiplication remains decidable when one adds to the language the usual order restricted to the prime numbers. We see moreover that the complexity of the latter theory is a tower of exponentials, of height O(n).

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

It is well known that if M is a module with finite spanning dimension, then one can talk of Sd(K), the spanning dimension of K only when K is a supplement submodule in M. In this paper we extend this concept to general submodules and obtained some important results. We characterize the set of all supplement submodules of the module R/(x) over R where R is a Euclidean domain and x ∈ R. Moreover, it is proved that the number of distinct supplements in R/(x) is 2k and Sd(R/(x)) = k where k is the number of distinct nonassociate prime factors of x.