2024/12/03 by Donald Silberger, Silberger, Donald
Mathematics · Physics and Astronomy · #Analytic Number Theory Research #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications
paper · pdf · doi:10.48550/arxiv.2412.02049
Let n,d, and k be positive integers where n and d are coprime. Our two main results are Theorem 1. There is a partition of the infinite interval [kd,∞) of positive integers into a family of finite sets X for which the sum of the reciprocals of the elements in X is n/d. Theorem 1. There is a partition of [2kd,∞) into an infinite family of infinite sets Y for which the sum of the reciprocals of the elements in Y is n/d. Our method is grounded in the Vital Identity, 1/z = 1/(z+1) + 1/z(z+1), which holds for every complex number z ∉ \-1,0\, and which gives rise to an eponymous algorithm that serves as our tool. At the core of our Theorems 1 and 2 is the number theoretic function ⋆: x↦ ⋆ x := x(x+1) into whose properties this paper continues an investigation initiated in [7].