Shells, sequences and intersections

Intersections of Recurrence Sequences

We derive sharp upper bounds for the size of the intersection of certain linear recurrence sequences. As a consequence of these, we partially resolve a conjecture of Yuan on simultaneous Pellian equations, under the condition that one of the parameters involved is suitably large.

Intersections of Second - Order Linear Recursive Sequences

Primes in Intersections of Beatty Sequences

In this note we consider the question of whether there are infinitely many primes in the intersection of two or more Beatty sequences ⌊ξjn+ ηj⌋, n ∈ N, j = 1, . . . , k. We begin with a straightforward sufficient condition for a set of Beatty sequences to contain infinitely many primes in their intersection. We then consider two sequences when one ξj is rational. However, the main result we est...

On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences

A Sierpiński number is an odd integer k with the property that k ·2+1 is composite for all positive integer values of n. A Riesel number is defined similarly; the only difference is that k · 2 − 1 is composite for all positive integer values of n. In this paper we find Sierpiński and Riesel numbers among the terms of the wellknown Fibonacci sequence. These numbers are smaller than all previousl...

Intersections of Descending Sequences of Affinely Equivalent Convex Bodies

Here it is shown that a set in Euclidean space can be represented as the intersection of a descending sequence of sets affinely equivalent to a given convex body, or arbitrarily closely approximated from above by sets affinely equivalent to the body, if and only if it is affinely equivalent to an affine retract of the body. For the special case in which the body is a simplex, the statement conc...