Improved Bounds on the Sizes of S.P Numbers

TL;DR

This paper improves the bounds on the sizes of S.P numbers, especially providing the first sharp bound for base 2, by refining previous methods and extending their applicability.

Contribution

It introduces an improved bound on the number of digits in S.P numbers, notably achieving the first sharp bound for base 2, based on modifications of Shah Ali's method.

Findings

First sharp bound for base-2 S.P numbers

Enhanced bounds for the size of S.P numbers in various bases

Extension of Shah Ali's method to obtain tighter bounds

Abstract

A number which is S.P in base r is a positive integer which is equal to the sum of its base-r digits multiplied by the product of its base-r digits. These numbers have been studied extensively in The Mathematical Gazette. Recently, Shah Ali obtained the first effective bound on the sizes of S.P numbers. Modifying Shah Ali's method, we obtain an improved bound on the number of digits in a base-r S.P number. Our bound is the first sharp bound found for the case r=2.