Probability Distribution of m Binary n-tuples

Probability of Mutually Commuting N-Tuples in Some Classes of Compact Groups

On the distribution of M-tuples of B-numbers

In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i) . G.J. Rieger (1965) and T. Cochrane / R.E. Dressler (1987) established bounds for the number of pairs (n, n+h) , resp., triples (n, n+1, n+2) of B -numbers up to a large real param...

probability of mutually commuting n-tuples in some classes of compact groups

Orbits on n-tuples

A transitive (infinite) permutation group which has m orbits on ordered pairs of distinct points has at least mn−1 orbits on ordered n-tuples. This is best possible, and groups attaining the bound can be characterised.

Diophantine m-tuples for primes

In this paper, we show that if p is a prime and ifA = {a1, a2, . . . , am} is a set of positive integers with the property that aiaj +p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f(ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m ...