The American Journal of Applied Sciences
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TYPE
Original Research
PAGE NO.
17-20
10.37547/tajas/Volume07Issue01-03
OPEN ACCESS
SUBMITED
25 October 2024
ACCEPTED
30 December 2024
PUBLISHED
23 January 2025
VOLUME
Vol.07 Issue01 2025
CITATION
Tilagova Buvgilos, & Ibroxim Sattarov. (2025). Perfect numbers and
their formula. Euclid and Euler’s approach. Do odd perfect number
exist?. The American Journal of Applied Sciences, 7(01), 17
–
20.
https://doi.org/10.37547/tajas/Volume07Issue01-03
COPYRIGHT
© 2025 Original content from this work may be used under the
terms of the creative commons attributes 4.0 License.
Perfect numbers and
their formula. Euclid and
Euler’s approach. Do
odd perfect number
exist?
Tilagova Buvgilos
Teacher, Jizzakh State pedagogical university academic lyceum,
Uzbekistan
Ibroxim Sattarov
Student, Jizzakh State pedagogical university academic lyceum,
Uzbekistan
Abstract:
Perfect numbers, one of the fundamental
concepts of mathematics have been focus of
mathematicians attention science ancient times.
This article discusses the concept of perfect
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
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The American Journal of Applied Sciences
mathematical analysis.
Perfect numbers
are defined as numbers that are
equal to the sum of all their positive divisors
excluding themselves.
Definition
: For a positive number N to be a perfect
number, the following equation must be satisfied:
σ(
N)
−
N = N , there is
𝜎
(N)-the sum of all divisors
of N.
In addition, the sum of the divisors of a
perfectnumber doubles: (N)=2*N
Examples:
1.
6 : Divisors 1, 2, 3. Sum 1+2+3=6
2.
28 : Divisors 1, 2 , 4 , 7 , 14. Sum
1+2+4+7+14=28
Around 300 BC, Euclid demonstrated that if 2p-1 is
prime, then the number
(
2
p-1
) * (
2
p
-1
)
perfect.
The first four perfect numbers were the only
numbers known to ancient Greek mathematics2.
In mathematics, Nicomachus found 8128 a
thousand years ago3. At the same time, there are 20
even perfect numbers. Another condition for a
number to be even is that it must be divisible by a
prime number p and 2p-1. Here exactly p and 2p-1
must be prime4.
In the 18th century, Leonhard Euler delved deeper
into Euclid's famous formula about prime numbers
and, based on this formula, proved the following
result:
Given N, it consists of two parts:
(2p-1) and (2p-1) . If (2p-1) is a prime number, the
divisors of N are as follows:
1.The divisors of (2p-1) :
These are 1, 2, 22, … , 2p
-1 , with a total of p divisors.
2.The divisors of (2p-1) : since 2p-1 is a prime
number, its divisors are only 1 and (2p-1).
3.All divisors of N :
{ 1, 2, 22, ..., (2p-1), (2p - 1), 2*(2p- 1), (22)*(2p - 1),
..., (2p-1)*(2p - 1)}
The number of divisor is 2*p.
Division Summation Numbers.
Finding the sum of the divisors of numbers.
1-
group: {1, 2, 22,…
, 2p-1}
This group’s sum is calculated using the formula of
the geometric series:
S1=1+2+22+…+(2p
-1)=2p-1
2-group: {(2p-1), 2*(2p-1), (22)*(2p-
1), … ,(2p
-
1)*(2p-1)}
We write this group in general form:
S2=(2p-
1)*(1+2+22+…+2p
-1)
We have S1=2p-1
S2=(2p-1)*(2p-1)=(2p-1)2
Calculate the sum of all divisor
The sum of all divisors is equal to S1 and S2:
σ(N)=S1+S2=(2p
-1)*(2p-1)2
We ara compare
𝜎
(N) with 2*N:
In formula N=(2p-1)*(2p-1)
Then: 2*N=2*(2p-1)*(2p-1)=(2p)*(2p-1)
Let’s check it:
𝜎
(N)=2p-1+(2p-1)*(2p-1)
This is also equal to 2*N, that is σ(N)=2*N
Even perfect numbers correspond exclusively to
Euclid's formula, that is1:
N=(2p-1)*(2p-1)
Euler made a significant contribution to the theory
of perfect numbers. He mainly studied the
properties
of
perfect
numbers
and
the
mathematical operations related to them in greater
depth. In Euler's work Introductio in Analysin
Infinitorum, a lot of information about perfect
numbers is presented. Among Euler's important
ideas are studies on whether perfect numbers are
even or odd. Euler, in particular, provided a detailed
analysis of even perfect numbers. Another
important idea presented by Euler is that if 2p-1 is a
prime number, then (2p-1)*(2p-1) is a
perfect number. This forms the foundation of
Euler's research on perfect numbers2.
Moreover, Euler studied the existence of odd
perfect numbers. It is known that any natural
number can be expressed as a product of its prime
factors.
N= p1a * p2b * ... * pnz
The strength of this function lies in the fact that it
can be expressed as a product of its components. It
is a multiplicative function3.
𝜎
(N)=
𝜎
(p1a * p2b * ... * pnz)=
𝜎
(p1a) *
𝜎
(p2b) *
… *
𝜎
(pnz)
Such as,
𝜎
(20)=
𝜎
(22)*
𝜎
(5)=(1+2+4)*(1+5)=42
Euler proved that every even number adheres to
the Euclidean algorithm.
This Euclid-Euler theorem solved a 1600 years old
problem. If N is such an odd perfect number that
exists, it is (N)=2*N . We can write N other shape N=
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The American Journal of Applied Sciences
p1a * p2b * ... * pnz
𝜎
(N)=
𝜎
(p1a * p2b * ... * pnz)=
𝜎
(p1a) *
𝜎
(p2b) *
… *
𝜎
(pnz)=2*N
The point to note is that if a prime number has an
odd exponent, its sigma will be even:
𝜎
(7)=1+7=8=2*k
Always (odd number)+(odd number)=even number.
The point to note is that if a prime number has an
even exponent, its sigma will be odd:
𝜎
(72)=1+7+7=15=2*k-1
So, (p2*k-1)=2*k ,
𝜎
(p2*k)=2*k-1 . Thus, Euler's
concept came into being. On the right side of this
𝜎
(p1a) *
𝜎
(p2b) *
… *
𝜎
(pnz)=2*N formula, there is
2*N. This means that there should be only one
factor of two on the left side. Because if there were
two factors of two, it would become
𝜎
(p1a)
*
𝜎
(p2b)
*
…*
𝜎
(pnz)=4*k=2*N This shows that
(4*k=2*N) N is even.1
From this, it follows that only one of the sigmas can
be even:
N=(pj2*k-1) *(p1a * p2b * ... * pnz)
Then, Euler refined the formula and proved that it
takes the form of : N=(p4*k+1)*(M2) However, he
did not prove whether these numbers exist or not.
In 1644, Marn Mersenne examined numbers in the
form of Euclid's formula, found the first 11 values of
𝜋
, and claimed that these were the prime numbers.
2, 5, 7, 13, 17, 19, 31, 67
However, he acknowledged that he had not verified
whether large numbers like
267-1=147573953589676412927 were prime or
not.
Scientists continued their research. Most of them
started with the list of prime numbers proposed by
Mersenne. In their list, 67 was included. Edouard
Lucas proved that 267-1 is not a perfect number.
In 1952, Rafael Robinson created a computer
program to find Mersenne prime numbers. Within
ten months, the next five Mersenne primes and
their corresponding perfect numbers were found1.
.
T/r
Mersenne prime
number
Perfect number
13
2
521
-1
(2
521
-1)* 2
521
14
2
607
-1
(2
607
-1)* 2
607
15
2
1279
-1
(2
1279
-1)*2
1279
16
2
2203
-1
(2
2203
-1)* 2
2203
17
2
2281
-1
(2
2281
-1)* 2
2281
Continuously, Mersenne prime numbers were found through the computer.
T/r
Mersenne prime numbers
Number of rooms
18
2
3217
-1
969
19
2
4253
-1
1281
20
2
4423
-1
1332
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2
9283
-1
2917
22
2
9941
-1
2993
23
2
11213
-1
3376
In 2017, scientists discovered the 50th Mersenne
prime number. It was the largest known Mersenne
prime at that time. In 2018, the 51st Mersenne
prime number, M82589933=282589933-1 was
discovered. It is currently the largest known
Mersenne prime.
Infinite perfect numbers:
If there are infinite prime numbers, then there are
also infinite perfect numbers.
CONCLUSION
Perfect numbers are an important topic that
reminds us of the contributions of Euclid and Euler
to mathematics. Their work laid the foundation for
many researches in mathematical analysis, algebra,
and number theory. Even today, mathematicians
are striving to uncover the mysteries of perfect
numbers. Mersenne prime numbers are so large
and rare that finding them requires a lot of time and
computer resources. In 1991, scientists determined
that if an odd perfect number exists, it must be
greater than 10300. New achievements have since
increased this number to 10220 . Given how large
the numbers are, it is unlikely that a computer will
find them anytime soon.
REFERENCES
Euclid’s “Elements” 9
-chapter
Euler, Leonhard, Introductio in Analysin Infinitorum
(1748)
A. Baker, A Concise Introduction to the Theory of
Numbers, Cambridge University Press, 1984.
Heath, Sir Thomas. A History of Greek Mathematics.
(Dover Publications, 1981). Chapter 6
David M. Burton Elementary Number Theory 2010,
7th edition
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