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We use the Bateman--Horn Conjecture from number theory to give strong evidence of a positive answer to Peter Neumann's question, whether there are infinitely many simple groups of order a product of six primes. (Those with fewer than six were classified by Burnside, Frobenius and Hölder in the 1890s.) The groups satisfying this condition are PSL2(8), PSL2(9) and PSL2(p) for primes p such that p² -- 1 is a product of six primes. The conjecture suggests that there are infinitely many such primes p, by providing heuristic estimates for their distribution which agree closely with evidence from computer searches. We also briefly discuss the applications of this conjecture to other problems in group theory, such as the classifications of permutation groups and of linear groups of prime degree, the structure of the power graph of a finite simple group, the construction of highly symmetric block designs, and the possible existence of infinitely many Kn groups for each n ≥ 5. [ABSTRACT FROM AUTHOR]

The Sandor-Smarandache function, denoted by SS(n), is a newly-introduced Smarandache-type arithmetic function. This paper focuses on the functions SS(30p), SS(60p), SS(210p), SS(420p) and SS(840p), where p (≥2) is a prime. At the end of the paper, four tables, giving the values of SS(30p), SS(60p), SS(210p) and SS(420p) for the first 200 primes, calculated on a computer, are given. [ABSTRACT FROM AUTHOR]

In this paper, we prove that, for any given odd number , the number of integers such that is a square is , where is the Euler totient function. Beyond this, we pose a conjecture and a problem for further research. [ABSTRACT FROM AUTHOR]

For any integer n ≥ 2, let P( n) be the largest prime factor of n. In this paper, we prove that the number of primes p ≤ x with P( p−1) ≥ p is more than (1− c+ o(1)) π( x) for 0 < c < 1/2. This extends a recent result of Luca, Menares and Madariaga for 1/4 ≤ c ≤ 1/2. We also pose two conjectures for further research. [ABSTRACT FROM AUTHOR]

In this paper we show that there exists a 3-dimensional continuum with two different decompositions into a direct product of prime factors in the shape category. This gives the positive answer to a question of K. Borsuk from his monograph “Theory of Shape” (1975). [ABSTRACT FROM AUTHOR]

A prime number p is called strong if p − 1 and p + 1 each have a large prime factor. Let x be a large positive number. We show a lower bound on the number of strong primes p between 1 and x such that p−1 and p+1 each have a prime factor larger than $${\sqrt{p}}$$ . [ABSTRACT FROM AUTHOR]

We investigate the distribution of integers with a fixed number of prime factors in arithmetic progressions, and obtain a generalization of the Siegel-Walfisz theorem under the extended Riemann hypothesis. As an application, we consider a problem of P. Erdős, A. M. Odlyzko and A. Sárközy about the representation of residue classes modulo q by products of two integers with a fixed number of prime factors. We show some conditional results. [ABSTRACT FROM AUTHOR]

The factorial-additive optimality of primes, i.e., that the sum of prime factors is always minimum, implies that prime numbers are a solution to an integer linear programming (ILP) encoding optimization problem. The summative optimality of primes follows from Goldbach's conjecture, and is viewed as an upper efficiency limit for encoding any integer with the fewest possible additions. A consequence of the above is that primes optimally encode—multiplicatively and additively—all integers. Thus, the set P of primes is the unique, irreducible subset of ℤ—in cardinality and values—that optimally encodes all numbers in ℤ, in a factorial and summative sense. Based on these dual irreducibility/optimality properties of P, we conclude that primes are characterized by a universal "quantum type" encoding optimality that also extends to non-integers. [ABSTRACT FROM AUTHOR]

Abstract: Kirchberg introduced weak exactness for von Neumann algebras as an analogue of exactness for -algebras. Ozawa found useful characterizations of weak exactness and he proved that weak exactness for group von Neumann algebras is equivalent to exactness of the groups. We generalize this concept to inclusions of -algebras in von Neumann algebras and study some fundamental properties and relationship to Kirchbergʼs weak exactness. We then give some permanence properties which are similar to those of exact groups. In the last section, we study a similar condition to Ozawaʼs condition (AO) with our weak exactness and generalize Ozawaʼs theorem for bi-exact groups. As a corollary, we give new examples of prime factors. [Copyright &y& Elsevier]

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]

Abstract: As the main result of this paper, we prove that there exists a continuum with non-trivial shape without any prime factor. This answers a question of K. Borsuk [K. Borsuk, Concerning the notion of the shape of compacta, in: Proc. Internat. Symposium on Topology and Its Applications, Herceg-Novi, 1968, pp. 98–104]. We also show that for each integer there exists a continuum X such that , but . Therefore we obtain the negative answer to another question of K. Borsuk [K. Borsuk, Some remarks concerning the shape of pointed compacta, Fund. Math. 67 (1970) 221–240]: Does , for a compactum X and some integer , implie that ? [Copyright &y& Elsevier]

Abstract: We answer a question of K. Borsuk (1972) showing that there exists a connected polyhedron of dimension 4 with two different decompositions into a direct product of prime factors (in the homotopy or shape category). [Copyright &y& Elsevier]

Let Fm = 2²m + 1 be the m'th Fermat number and Fm = p1p2 ... pl with primes p1 ≤ p2 ≤ ... ≤ pl. In this note, we derive some restricted relations for the prime factors. For example, we prove that pl + pl-1 + ... + pl-t+1 ≥ (2m+2(4m + 8) + 1)t holds for any positive integer t. [ABSTRACT FROM AUTHOR]