Parasitic Numbers at Arbitrary Base

TL;DR

This paper develops a general formula to generate parasitic numbers in any base, extending known results from decimal to other number systems, and demonstrates its application in multiple bases.

Contribution

A new analytical formula for generating parasitic numbers in arbitrary bases, not relying on their cyclical properties, is derived and demonstrated.

Findings

Formula successfully generates parasitic numbers in bases 3, 4, 5, 8, 10, and 16.

The approach extends known decimal results to other bases.

The method allows control over the number of periods in generated parasitic numbers.

Abstract

A natural number is called an {\lambda}-parasitic number if it is multiplied by integer {\lambda} as the rightmost digit moves to the front. The Full set of these numbers is known in the decimal system. Here, a formula to analytically generate parasitic numbers in any base was derived and demonstrated for the number systems in base t = 3, 4, 5, 8, 10 and 16. It allows to generate parasitic numbers with given numbers of periods. The formula was derived using the definition of parasitic numbers , not their cyclical property.