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The mathematical magic of perfect numbers.

Article Excerpt
ABSTRACT

Mathematicians have been fascinated for centuries by the properties and patterns of numbers [2]. 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 [2]. In this paper, we will investigate some important properties of perfect numbers. We give easy and simple proofs of theorems such that students with calculus II skills can understand most of it. 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..

Key Words: Prime Numbers, Perfect numbers, and Triangular numbers.

INTRODUCTION AND BACKGROUND

Throughout history, there have been studies on perfect numbers. It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity [3]. It is rather likely, although not completely certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, where detailed justification for this idea is given [2]. Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties [1]. Although, the four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries [5]. The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's Elements written around 300BC [3].

THE MAIN RESULTS

Proposition I: If [2.sup.n]-1 is prime, then...

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