NOTE ON THE TWO CELEBRITY NUMBERS, PERFECT NUMBERS AND TRIANGULAR NUMBERS

Abstract

Mathematicians have been fascinated for centuries by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. Thus a positive integer is called a perfect number if it is equal to the sum of its proper positive divisors. The search for perfect numbers began in ancient times. The four perfect numbers 6, 28, 496, and 8128 seem to have been known from ancient times. In this paper, we will investigate some important properties of perfect numbers. We give easy and simple proofs of theorems using finite series. We give our own alternative proof of the well-known Euclid’s Theorem

(Theorem I). We will also prove some important theorems which play key roles in the mathematical theory of perfect numbers.