A combinatorial problem and congruences for the Rayleigh function

Abstract

Let z be a positive integer and let m be the number of nonzero terms in the base 2 expansion of z. Define /(z, s) as the number of positive integers rgz/2 such that the number of nonzero terms in the base 2 expansion of r plus the number of nonzero terms in the base 2 expansion of z-r is equal to m+s. We find formulas for f(z, s) and show how these formulas can be used in proving congruences for the Rayleigh function.

A Property of a Class of Nonlinear Difference Equations

“…This example is discussed in [4] and [3]. We note that Theorem 4.1 proves a conjecture in [4], namely that if b=(2k+l)2t, t>0, then 29Mo2n(alb) m (-l)°((2/c + l)/a) (mod 4).…”

“…We note that one of these examples is discussed in [4] and [3] and that a conjecture in [4] is proved in this paper.…”

“…The following lemmas are proved in [3] and are necessary for the proofs of the congruences in this paper. …”

The number of binomial coefficients divisible by a fixed power of 2

“…In this paper we consider the same problem for ¿ = 2. We note that this problem is also discussed briefly in [3].…”

“…Then we have We note that the formulas for/(n, 0) and f(n, 1) can be found in [1] and [3]. In all of the following theorems, we assume that (1.1) holds and fin, j) is given by (2.2).…”