Let Δn = {(x, y): x, y are integers 1 ≤ x, y ≤ n} be the n x n square array of integer lattice points in the plane.

In this note we introduce recurring-with-carry sequences which generalize add-with-carry and subtract-with-borrow sequences introduced in [1], and describe periods of admissible recurring-with-carry sequences which include add-with-carry and subtract-with-borrow sequences with a few exceptions.

This article studies particular sequences satisfying polynomial recurrences, among those Apéry's sequence which is shown to be the Legendre transform of the sequence. This results in the construction of simultaneous approximations of π 2/8 and ζ(3).

We consider the Egyptian fraction equation and discuss techniques for generating solutions. By examining a quadratic recurrence relation modulo a family of primes we have found some 500 new infinite sequences of solutions. We also initiate an investigation of the randomness of the distribution of solutions, and show that there are infinitely many solutions not generated by the aforementioned technique.

We prove a result related to the Erdős-Ginzburg-Ziv theorem: Let p and q be primes, α a positive integer, and m∈{pα, pαq}. Then for any sequence of integers c= {c1, c2,…, cn} there are at least

subsequences of length m, whose terms add up to 0 modulo m (Theorem 8). We also show why it is unlikely that the result is true for any m not of the form pα or pαq (Theorem 9).

Three differently defined classes of two-symbol sequences, which we call the two-distance sequences, the linear sequences and the characteristic sequences, have been discussed by a number of authors and some equivalences between them are known. We present a self-contained proof that the three classes are the same (when ambiguous cases of linear sequences are suitably in terpreted). Associated with each sequence is a real invariant (having a different appropriate definition for each of the three classes). We give results on the relation between sequences with the same invariant and on the symmetry of the sequences. The sequences are closely related to Beatty sequences and occur as digitized straight lines and quasicrystals. They also provide examples of minimal word proliferation in formal languages.

§1. Introduction. In 1985, Sárkõzy [11] proved a conjecture of Erdõs [2] by showing that the greatest square factor s(n)2 of the “middle” binomial coefficient satisfies for arbitrary ε > 0 and sufficiently large n

Where

Let L be a linear differential operator with rational coefficients such that 0 is not an irregular singularity of L and that for sufficiently many p's the equation Lv = 0 has no zero solution mod p. We show that if u is a formal power series whose coefficients are p-adic integers for almost all p and if Lu is rational, then u too is rational.