Mathematics Subject Classification (2000): Primary 11M26; Secondary 11K38 We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to Montgomery’s pair correlation function and the distribution of primes in short...

Let 〈P〉 ⊂ N be a multiplicative subsemigroup of the natural numbers N= {1, 2, 3, . . .} generated by an arbitrary set P of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈P〉:n≤x (μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈P (1− (1/p)) (the case where P is all the primes is a we...

Let X be a definable C manifold, Y1, Y2 definably compact definable C r submanifolds of X such that dimY1 + dimY2 < dimX and Y1 has a trivial normal bundle. We prove that there exists a definable isotopy {hp}p∈J such that h0 = idX and h1(Y1) ∩ Y2 = ∅. 2010 Mathematics Subject Classification. 14P10, 14P20, 58A05, 03C64.

This is a survey article on primality testing and general methods for recognizing primes. We also look at methods for generating random primes since this is valuable in cryptographic methods such as the RSA cipher. Mathematics Subject Classification: Primary: 11A41; Secondary: 11A51

In this note, we mainly obtain the equation x2m − yn = z2 have finite positive integer solutions (x, y, z,m, n) satisfying x > y be two consecutive primes. Mathematics Subject Classification: 11A41; 11D41

In this paper, we give some new examples of polynomial D(n)triples and quadruples, i.e. sets of polynomials with integer coefficients, such that the product of any two of them plus a polynomial n ∈ Z[X] is a square of a polynomial with integer coefficients. The examples illustrate various theoretical properties and constructions for a quadratic polynomial n which appeared in recent papers. One ...

A prime Pythagorean triangle has three integer sides of which the hypotenuse and one leg are primes. In this article we investigate their properties and distribution. We are also interested in finding chains of such triangles, where the hypotenuse of one triangle is the leg of the next in the sequence. We exhibit a chain of seven prime Pythagorean triangles and we include a brief discussion of ...

This paper proves that if N is a nonnegative eligible integer, coprime to 7, which is not of the form x2+y2+7z2, thenN is square-free. The proof is modelled on that of a similar theorem by Ono and Soundararajan, in which relations between the number of representations of an integer np2 by two quadratic forms in the same genus, the pth coefficient of an L-function of a suitable elliptic curve, a...

Weuse a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field (exp(2π i/m)) has an infinite Hilbert p-class field tower with high rankGalois groups at each step, simultaneously for all primes p of size up to about (log logm)1+o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower ove...

A modification of Katzman’s example is given to produce a two-generated ideal in a two-dimensional Noetherian integral domain for which the set of associated primes of all the Frobenius powers is infinite. A further modification yields a four-dimensional Noetherian integral domain and a five-dimensional Noetherian local integral domain for which an explicit second local cohomology module has in...