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In this paper, we characterize the odd positive integers n satisfying the congruence $$\sum\nolimits_{j = 1}^{n - 1} {j^{\tfrac{{n - 1}} {2}} } \equiv 0 (mod n)$$. We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8. [ABSTRACT FROM AUTHOR]

Let P denote an almost-prime with at most r prime factors, counted according to multiplicity, and let E( N) denote the number of natural numbers not exceeding N that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one P. Then we have E( N)≪log N. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with P in place of P. [ABSTRACT FROM AUTHOR]

The article focuses on a property of positive integers depicting the interpedendence between their construction and the form of their prime factors as established by E. A. Karatsuba and M. E. Changa. Karatsuba has discovered that the deviation of the cardinalities of sets of numbers with an odd and an even number of prime factors has the tendency to reach infinity under certain conditions on the factors. It presents several theorems which constitute the distribution of positive integers whose prime factors belong to certain subsets.

Let N be a sufficiently large even integer. Let p denote a prime and P denote an almost prime with at most two prime factors. In this paper, it is proved that the equation N = p + P ( p ≤ N) is solvable. [ABSTRACT FROM AUTHOR]

recent result, conjectured by Arnold and proved by Zarelua, states that for a prime number p, a positive integer k, and a square matrix A with integral entries one has $${\textrm tr}(A^{p^k}) \equiv {\textrm tr}(A^{p^{k-1}}) ({\textrm mod}{p^k})$$. We give a short proof of a more general result, which states that if the characteristic polynomials of two integral matrices A, B are congruent modulo p then the characteristic polynomials of A and B are congruent modulo p, and then we show that Arnold's conjecture follows from it easily. Using this result, we prove the following generalization of Euler's theorem for any 2 × 2 integral matrix A: the characteristic polynomials of A and A are congruent modulo n. Here ϕ is the Euler function, $$\prod_{i=1}^{l} p_i^{\alpha_i}$$ is a prime factorization of n and $$\Phi(n)=(\phi(n)+\prod_{i=1}^{l} p_i^{\alpha_i-1}(p_i+1))/2$$. [ABSTRACT FROM AUTHOR]

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by $\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}$ , we show that for any given α > 0 the interval $(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]$ contains an integer having no prime factor exceeding X for all X sufficiently large. [ABSTRACT FROM AUTHOR]

In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + ε)/ log 2) log log n and generalize it to the equation a! a! ... a! = n!. In a recent paper, F. Luca proved that n − b = 1 for large enough n provided that the ABC-hypothesis holds. [ABSTRACT FROM AUTHOR]

We prove that whenever $$ \mathcal{A} $$ and $$ \mathcal{B} $$ are dense enough subsets of {1, ..., N}, there exist a ∈ $$ \mathcal{A} $$ and b ∈ $$ \mathcal{B} $$ such that the greatest prime factor of ab + 1 is at least $$ N^{1 + |\mathcal{A}|/(9N)} $$. [ABSTRACT FROM AUTHOR]

Let p denote a prime and P 2 denote an almost prime with at most two prime factors. The author proves that for sufficiently large x, [Figure not available: see fulltext.] where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu. [ABSTRACT FROM AUTHOR]

We study the irrational factor function I( n) introduced by Atanassov and defined by $$ I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } } $$ , where $$ n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } } $$ is the prime factorization of n. We show that the sequence { G( n)/ n} n≧1, where G( n) = Π I( ν)1/ n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I( n). [ABSTRACT FROM AUTHOR]

Abstract  Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I 2 ∉ A (here I 2 is the two-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A ⊆ Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD. [ABSTRACT FROM AUTHOR]

Let S be a bielliptic surface over a finite field, and let the elliptic curve B be the image of the Albanese mapping S → B. In this case, the zeta function of the surface is equal to the zeta function of the direct product ℙ1 × B. A classification of the possible zeta functions of bielliptic surfaces is also presented in the paper. [ABSTRACT FROM AUTHOR]

Abstract  Algorithms are proposed that construct the basis of the set of solutions to a system of homogeneous or inhomogeneous linear Diophantine equations in a residue ring modulo n when the prime factors of n are known. [ABSTRACT FROM AUTHOR]

We disprove some power sum conjectures of Turán that would have implied the density hypothesis of the Riemann zeta-function if true. [ABSTRACT FROM AUTHOR]

The main result of this paper is the fact that the fraction of primes p ≤ x satisfying the condition that p − 1 has a prime divisor q > exp(ln x/ln ln x) and the number of prime divisors of q − 1 essentially differ from ln ln( x/n), where n = ( p − 1)/ q, tends to zero as x increases. [ABSTRACT FROM AUTHOR]

The abc–conjecture for the ring of integers states that, for every ε > 0 and every triple of relatively prime nonzero integers ( a, b, c) satisfying a + b = c, we have max(| a|, | b|, | c|) ≤ rad( abc)1 + ε with a finite number of exceptions. Here the radical rad( m) is the product of all distinct prime factors of m. In the present paper we propose an abc–conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number. From this abc–conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over ℚ. [ABSTRACT FROM AUTHOR]

The article considers a certain class of homeomorphisms of closed, oriented 3-manifolds that have a connected sum decomposition into prime factors. It discusses the definition of slide homeomorphism. It shows that away from the sliding set, f can be isotoped to a fixed point free homeomorphism by an arbitrary small isotopy. It considers fixed points over the sliding set and argues that f is isotopic to a homeomorphism with finitely many fixed points.

For a finite group G, let τ( G) denote a set of primes such that a prime p belongs to τ( G) if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions: (1) G has a p-complement for each p∈τ( G); (2) ∣τ( G)∣=2; (3) the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈τ( G); then G either is solvable or G/Sol( G)≅(2, 7) where Sol( G) is the largest solvable normal subgroup of ( G). [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper we prove that all tetravalent connected Cayley graphs of order p3 are normal. As an application, a classification of tetravalent symmetric graphs of odd prime–cube order is given. [ABSTRACT FROM AUTHOR]

We introduce a new class of lightlike submanifolds, namely, Screen Cauchy Riemann (SCR) lightlike submanifolds of indefinite Kaehler manifolds. Contrary to CR-lightlike submanifolds, we show that SCR-lightlike submanifolds include invariant (complex) and screen real subcases of lightlike submanifolds. We study some properties of proper totally umbilical SCR-lightlike submanifolds, their invariant (complex) and screen real subcases. [ABSTRACT FROM AUTHOR]

Let N be a sufficiently large even integer. In this paper it is proved that the equation.N = p+P[sub 2] , p ≤ N[sup 0,95] ,.is solvable, where p denotes a prime and P[sub 2] denotes an almost prime with at most two prime factors. [ABSTRACT FROM AUTHOR]

Let d[sub k](n) denote the k-fold iterated divisor function (k ≥ 2). It is proved that for sufficiently large x, d[sub k](n) = d[sub k](n + 1) holds for ≫ x(log log x)[sup -3] integers n ≤ x. [ABSTRACT FROM AUTHOR]

Abstract. One of the earliest applications of Cantor's Normal Form Theorem is Jacobstahl's proof of the existence of prime factorizations of ordinals. Applying the techniques of reverse mathematics, we show that the full strength of the Normal Form Theorem is used in this proof. [ABSTRACT FROM AUTHOR]