Nargiza Akmalova
By: Nargiza Akmalova
As soon as the student goes to school, from the 1st grade, he is taught numbers and numbers in mathematics classes. Every time it is shown step by step to higher classes, the types and properties of numbers are taught. Students learn to differentiate between positive numbers, negative numbers, prime numbers, complex numbers and perfect numbers. In this article, we will discuss in detail about perfect numbers.
Perfect numbers are natural numbers that are equal to the sum of their natural divisors. For example, the smallest perfect number is 6. We know that the natural divisors of 6 are numbers 1, 2 and 3. According to the definition of perfect numbers, the number 6 satisfies the equality 6=1+2+3. The second perfect number is 28. Because it is also equal to the sum of its divisors: 28=14+7+4+2+1.
A perfect sequence of 6
28
496
8128
33550336
8589869056
137438691328
2305843008138952128
2658455991569831744654692615953842176
191561942608236107294793378084303638130997321548169
………….
Natural numbers are increasing in the number axis, perfect numbers are decreasing. All the perfect numbers identified so far are even numbers, and no odd perfect numbers have yet been identified. However, the non-existence of odd perfect numbers has not been mathematically proven. It has not yet been proven that the perfect numbers are infinite. Because to prove this and to find these infinitely complex numbers, not even the smartest mathematicians could do it. Euclid was the first to describe the method of determining the perfect number, in modern language, the algorithm of determining the perfect number. In Volume IX of his work “Fundamentals”, if the expression 2^(p-1 ) is a prime number, then 2^(p-1 ) (2^(p )-1) is a perfect even number was mentioned. Later, Leonard Euler proved that all even perfect numbers obey the above expression given by Euclid.
In the late 19th and early 20th centuries, mathematicians discovered three more perfect numbers corresponding to p=89, p=107, and p=127. But these were the last perfect numbers found by human resources and patience. Because now we are not talking about the room of billions or trillions, but about even larger numbers. All the perfect numbers that have been found after that have been found by computers. The largest perfect number known to date is 2^(43112609)-1, which consists of 12978189 digits. Research is also being conducted to find perfect numbers larger than this. But to find such a large number, even the most powerful computers of today will need to work continuously for years. Mathematicians hope that in the near future quantum computers will be able to solve these problems.
Nargiza Akmalova was born on September 5, 2005 in Tashkent. Graduated from school in 2020 with an honors certificate. In 2022, he became the recipient of the “O’zbekiston Milliy Universiteti faxri” scholarship and graduated from the republican academic lyceum named after S.H. Sirojiddinov with an excellent diploma. In the same year, Tashkent State Pedagogical University named after Nizami was admitted to the department of Mathematics and Informatics. Currently a 2nd year student. In 2023, she became the owner of the state scholarship named after “Abdulla Avloniy”. Akmalova Nargiza is the author of more than 15 articles and the owner of more than 60 international certificates. She is a member of a number of international organizations. In particular, “Qo’sh qanot” Association of Writers and Creators of Kazakhstan, Including Global education ambassador, SPSC ambassador, Argentina’s Juntos por las letas, Global education ambassador of Wisdom University, All India Council For Technical Skill Development, Global Fr
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