A binomial representation of the 3x + 1 problem
Let the collection of arithmetic sequences be a disjoint covering system of the integers. We prove that if for some primes and integers , then there is a such that . We conjecture that the divisibility result holds for all moduli.A disjoint covering system is called saturated if the sum of the reciprocals of the moduli is equal to . The above conjecture holds for saturated systems with such that the product of its prime factors is at most .
A number of properties of a function which originally appeared in a problem proposed by Ramanujan are presented. Several equivalent representations of the function are derived. These can be used to evaluate the function. A new derivation of an expansion in inverse powers of the argument of the function is obtained, as well as rational expressions for higher order coefficients.
Let be a system of disjoint subsets of . In this paper we examine the existence of an increasing sequence of natural numbers, , that is an asymptotic basis of all infinite elements of simultaneously, satisfying certain conditions on the rate of growth of the number of representations , for all sufficiently large and A theorem of P. Erdös is generalized.
Let m ≥ 2 be a positive integer. Given a set E(ω) ⊆ ℕ we define to be the number of ways to represent N ∈ ℤ as a combination of sums and differences of m distinct elements of E(ω). In this paper, we prove the existence of a “thick” set E(ω) and a positive constant K such that for all N ∈ ℤ. This is a generalization of a known theorem by Erdős and Rényi. We also apply our results to harmonic analysis, where we prove the existence of certain thin sets.