Perfect numbers and their formula. Euclid and Euler’s approach. Do odd perfect number exist?

Аннотация

Perfect numbers, one of the fundamental concepts of mathematics have been focus of COPYRIGHT mathematicians attention science ancient times. © 2025 Original content from this work may be used under the This article discusses the concept of perfect terms of the creative commons attributes 4.0 License. numbers, their identification formulas, the Euclid and Euler approaches, as well as of the most debated problems the existence of odd perfect numbers. Keywords: Perfect numbers, fundamental concepts Introduction: Perfect numbers are a mysterious aspect of mathematical beauty. Throughout the history of mathematics, perfect numbers have held a special place. Euclid, in his famous work Elements, outlined the principles of the formation of perfect numbers, while Euler expanded this theory further and established a solid scientific foundation for the role of perfect numbers in modern mathematics. Additionally, Mersenne prime numbers play a significant role in the formation of perfect numbers. To this day, all identified perfect numbers are even, and the question of the existence of odd perfect numbers remains open. Perfect numbers are not only important for theoretical mathematics but also hold significance for the overall development of number theory. Therefore, studying this topic in greater depth is beneficial not only from a historical perspective but also from the standpoint of modern The American Journal of Applied Sciences 17 https://www.theamericanjournals.com/index.php/tajas The American Journal of Applied Sciences mathematical analysis. 1)*(2p-1)} We write this group in general form: Perfect numbers are defined as numbers that are S2=(2p-1)*(1+2+22+…+2p-1) equal to the sum of all their positive divisors We have S1=2p-1 excluding themselves. S2=(2p-1)*(2p-1)=(2p-1)2 Definition: For a positive number N to be a perfect Calculate the sum of all divisor number, the following equation must be satisfied: The sum of all divisors is equal to S1 and S2: σ(N) − N = N , there is 𝜎(N)-the sum of all divisors σ(N)=S1+S2=(2p-1)*(2p-1)2 of N. We ara compare 𝜎(N) with 2*N: In addition, the sum of the divisors of a perfectnumber doubles: (N)=2*N In formula N=(2p-1)*(2p-1) Examples: Then: 2*N=2*(2p-1)*(2p-1)=(2p)*(2p-1) 1. 6 : Divisors 1, 2, 3. Sum 1+2+3=6 Let’s check it: 2. 28 : Divisors 1, 2 , 4 , 7 , 14. Sum 𝜎(N)=2p-1+(2p-1)*(2p-1) 1+2+4+7+14=28 This is also equal to 2*N, that is σ(N)=2*N Around 300 BC, Euclid demonstrated that if 2p-1 is Even perfect numbers correspond exclusively to prime, then the number (2p-1) * (2p-1) perfect. Euclid's formula, that is1: The first four perfect numbers were the only N=(2p-1)*(2p-1) numbers known to ancient Greek mathematics2. Euler made a significant contribution to the theory In mathematics, Nicomachus found 8128 a of perfect numbers. He mainly studied the thousand years ago3. At the same time, there are 20 properties of perfect numbers and the even perfect numbers. Another condition for a mathematical operations related to them in greater number to be even is that it must be divisible by a depth. In Euler's work Introductio in Analysin prime number p and 2p-1. Here exactly p and 2p-1 Infinitorum, a lot of information about perfect must be prime4. numbers is presented. Among Euler's important In the 18th century, Leonhard Euler delved deeper ideas are studies on whether perfect numbers are into Euclid's famous formula about prime numbers even or odd. Euler, in particular, provided a detailed and, based on this formula, proved the following analysis of even perfect numbers. Another result: important idea presented by Euler is that if 2p-1 is a prime number, then (2p-1)*(2p-1) is a Given N, it consists of two parts: perfect number. This forms the foundation of (2p-1) and (2p-1) . If (2p-1) is a prime number, the Euler's research on perfect numbers2. divisors of N are as follows: Moreover, Euler studied the existence of odd 1.The divisors of (2p-1) : perfect numbers. It is known that any natural These are 1, 2, 22, … , 2p-1 , with a total of p divisors. n u m b e r c a n b e e x p r e s s e d a s a p r o d u c t o f i t s prime 2.The divisors of (2p-1) : since 2p-1 is a prime factors. number, its divisors are only 1 and (2p-1). N= p1a * p2b * ... * pnz 3.All divisors of N : T h e s t r e n g t h o f t h i s f u n c t i o n l i e s i n t h e f a c t that it { 1, 2, 22, ..., (2p-1), (2p - 1), 2*(2p- 1), (22)*(2p - 1), can be expressed as a product of its components. It ..., (2p-1)*(2p - 1)} is a multiplicative function3. The number of divisor is 2*p. 𝜎(N)=𝜎(p1a * p2b * ... * pnz)=𝜎(p1a) * 𝜎(p2b) * … * 𝜎(pnz) Division Summation Numbers. Such as, Finding the sum of the divisors of numbers. 𝜎(20)=𝜎(22)*𝜎(5)=(1+2+4)*(1+5)=42 1-group: {1, 2, 22,…, 2p-1} Euler proved that every even number adheres to This group’s sum is calculated using the formula of the Euclidean algorithm. the geometric series: This Euclid-Euler theorem solved a 1600 years old S1=1+2+22+…+(2p-1)=2p-1 problem. If N is such an