1.1. Earlier work. Let ` be a prime. In our previous work [KLS], which generalised a result of Wiese [W], the Langlands functoriality principle was used to show that for every positive integer t there exists a positive integer k divisible by t such that the finite simple group Cn(`) = PSp2n(F`k) is the Galois group of an extension of Q unramified outside {∞, `, q} where q 6= 2 is a prime that d...

The initial motivation of the present paper is the following problem of Erd6s and Woods, whose solution would be of interest in logic (see [3, 6, 11, 10, 16]): Does there exist an integer k >=2 with the following property : i f x and y are positive integers such that for l<=i<=k, the two numbers x+ i and y+i have the same prime factors, then x=y. For each integer n->2, let us denote by Supp (n)...

Balanced generalized weighing matrices are applied for constructing a family of symmetric designs with parameters (1 + qr(rm+1 − 1)/(r − 1), rm, rm−1(r − 1)/q), where m is any positive integer and q and r = (qd − 1)/(q − 1) are prime powers, and a family of non-embeddable quasi-residual 2−((r+1)(rm+1−1)/(r−1), rm(r+ 1)/2, rm(r− 1)/2) designs, where m is any positive integer and r = 2d− 1, 3 · 2...

Almost integral TQFT was introduced by Gilmer [G]. For each simple Lie algebra g and some prime integer we associate an almost integral TQFT which derives the projective Witten-Reshetikhin-Turaev invariant τ for closed 3-manifolds. As a corollary, one can show that τ is an algebraic integer for certain prime integers. The result in this paper can be used to prove that τ M satisfies some Murasug...

A positive integer n is called a k-imperfect number if kρ(n) = n for some integer k ! 2, where ρ is a multiplicative arithmetic function defined by ρ(pa) = pa−pa−1+pa−2−· · ·+ (−1)a for a prime power pa. In this paper, we prove that every odd k-imperfect number greater than 1 must be divisible by a prime greater than 102, give all k-imperfect numbers less than 232 = 4294 967 296, and give sever...

1. If n is a positive integer, p is a prime number and k is a non-negative integer with p | n, p n then we write p‖n. For n > 1 let p(n) and P (n) denote the least and greatest prime factor of n, respectively. In the last 15 years many papers have been written on the arithmetical properties of elements of sum sets A + B (defined as the set of the integers of the form a + b with a ∈ A, b ∈ B) wh...

He noted then that the conjecture is true if n = p 1 2 or if n = p where p is a prime and r a positive integer. Calculations showed the conjecture also held for n 100. Recently, in a study of more general polynomials, the rst author [2] obtained further irreducibility results for f(x); in particular, he established irreducibility in the case that n+ 1 is a squarefree number 3 and in the case th...

For every even integerN , denote byD(N) andD1,2(N) the number of representations of N as a sum of two primes and as a sum of a prime and an integer having at most two prime factors, respectively. In this paper, we give a new upper bound for D(N) and a new lower bound for D1,2(N), which improve the corresponding results of Chen. We also obtain similar results for the twin prime problem

Yes, but such a formula is complicated. For example, is there a polynomial f ∈ Z[x] for which f(n) = pn? f(x) = anx n + · · ·+ a1x+ a0 f(a0) = ana0 n + · · ·+ a1a0 + a0 so a0 | f . Suppose q is prime and f(n) = q. Then q | f(n + kq) for each k ∈ Z. So, in particular, we see that if f(m) is prime for each positive integer m, then f is a constant. In particular, f(x) = q for some prime q. The pol...

0.1.3 Unique Factorization Theorem If a is an integer, not 0 or ±1, then (1) a can be written as a product p1 · · · pn of primes. (2) If a = p1 · · · pn = q1 · · · qm, where the pi and qj are prime, then n = m and, after renumbering, pi = ±qi for all i. [We allow negative primes, so that, for example, −17 is prime. This is consistent with the general definition of prime element in an integral d...