We are faced with numbers literally every moment of our earthly life. Even the ancient Greeks had gematria (numerology). For the image of numbers letters of the alphabet were used. Each name or written word corresponds to a specific number. Today, the science of mathematics has reached a very high degree of development. There are so many numbers used in various calculations that they are grouped into specific groups. A special place among them is occupied by perfect numbers.
The origins
In ancient Greece, people compared the properties of numbers according to their names. Divisors of numbers were assigned a special role in numerology. In this regard, the ideal (perfect) numbers were those that were equal to the sum of their divisors. But, the ancient Greeks did not include the number itself in the divisors. To better understand what perfect numbers are, let us show it with examples.
Based on this definition, the smallest ideal number is 6. After it will be 28. Then 496.
Pythagoras believed that there are special numbers. Euclid was of the same opinion. For them, these numbers were so unusual and specific that they associated them with mystical ones. Such numbers tend to be perfect. This is what perfect numbers are for Pythagoras and Euclid. These included 6 and 28.
Key
Mathematicians always strive to find a common key for finding an answer when solving a problem with several options.
So, they were looking for a formula that determines the ideal number. But only a hypothesis was obtained, which still needed to be proved. Imagine having already determined what perfect numbers are, mathematicians spent more than a thousand years to determine the fifth of them! After 1,500 years, it became known.
Scientists Fermat and Mersen (XVII century) made a very significant contribution to the calculation of ideal numbers. They proposed a formula for their calculation. Thanks to French mathematicians and the works of many other scientists, at the beginning of 2018, the number of perfect numbers reached 50.
Progress
Of course, if the discovery of the perfect number, which was already the fifth in a row, took one and a half millennia, today thanks to computers they are calculated much faster. For example, the opening of the 39th ideal number came in 2001. It has 4 million characters. In February 2008, the 44th perfect number was opened. In 2010, it was 47th ideal, and by 2018, as mentioned above, the 50th was opened with the status of excellence.
There is another interesting feature. Studying what perfect numbers are, mathematicians made a discovery - they are all even.
A bit of history
It is not known for certain when the numbers corresponding to the ideal were first noticed. However, it is believed that even in ancient Egypt and Babylon they were depicted on a finger account. And it’s easy to guess what perfect number they represented. Of course, this was 6. Until the fifth century AD, a score was kept with the fingers. To show the number 6 on the hand, the ring finger was bent and the others straightened.
In ancient Egypt, the elbow served as a measure of length. It was equivalent to twenty eight fingers. And, for example, in Ancient Rome there was an interesting custom - to take sixth place at feasts for honored and noble guests.
Followers of Pythagoras
The followers of Pythagoras, too, were fond of ideal numbers. Which of the numbers is perfect after 28, was very interested in Euclid (IV century BC). He gave the key to finding all the perfect even numbers. Of interest is the ninth book of the Euclidean "Beginnings." Among his theorems, there is one that explains that a number that has a wonderful property is called perfect:
the value of p will be equivalent to the expression 1 + 2 + 4 + ... + 2n, which can be written as 2n + 1-1. This is a prime number. But already 2np will be perfect.
To verify the validity of this statement, we need to consider all the proper divisors of 2np and calculate their sum.
This discovery is believed to belong to the disciples of Pythagoras.
Euclidean rule
In addition, Euclid proved: the form of an even perfect number is mathematically represented as 2n-1 (2n-1). If n is prime and 2n-1 will be prime.
The Euclidean rule was used by the ancient Greek mathematician Nicomachus of Gerasa (I-II c.). He found ideal numbers as 6, 28, 496, 8128. Nikomakh Gerazsky spoke about ideal numbers as about very beautiful, but small mathematical concepts.
One and a half thousand years later, the German scientist Regiomontan (Johann Muller) discovered the fifth perfect number in mathematics. They turned out to be 33,550,336.
Further search for mathematicians
The numbers, which are considered to be simple and belong to the 2n-1 series, are called Mersenne numbers. This name was given to them in honor of the French mathematician who lived in the 17th century. It was he who discovered the eighth perfect number in 1644.
After 250 years, the Russian scientist mathematician Pervushin I.M. from the Perm province found the ninth ideal number.
Since 1952, computers (electronic computers) were connected to such mathematical studies. The speed of calculations has increased significantly. For example, it became known that, unlike the first ideal number 6, which is unique, the twenty-fourth has more than 12,000 characters in its arsenal!
The story of the chessboard
There is one very interesting story about a chessboard, a king and grain. Once the king, being delighted with the game of chess, invited the creator of the game to choose a reward for himself. Then the sage chose a modest, seemingly reward - to put grain on the cells of a chessboard. I was surprised by the layout: on the first cell 1 grain, on the second - 2, the third cell should contain 4, and so fill the entire board. It is interesting that in the last 64 cells there were 1,199,038,364,791.120 tons, which is 18,446,744,073,709,709,551,615 grains.
This amount is approximately 1800 times higher than the world wheat crop collected in human history.
If we consider the mass of one grain as 0.065 g, then the total mass on the chessboard will be 1,200 trillion tons.
If it were necessary to build a barn to store such an amount of grain, then its size would be larger than Mount Everest: 10 x 10 x 15 (km), and in volumes it would be about 1,500 km³!
Numerology
In numerology, there is such a thing as the most perfect number 108, bringing success. Its roots go back to Vedic culture. It is believed that if you perform a certain action exactly 108 times, then in this event a certain level of perfection will be achieved. This opinion is associated with the structure of human memory: it is divided into short-term and permanent (internal). So, it is in the internal memory that those concepts that a person fulfilled 108 times are placed. Perhaps that is why the prayer beads in the classical version contain exactly 108 beads. So, after reading a prayer in a full circle of beads, it becomes part of a person’s permanent memory.
Mystery and Facts
To understand whether a number is perfect, certain calculations need to be done. There is no other way. And such numbers are rare. For example, the Pythagorean Yamblich wrote about ideal numbers as a phenomenon that occurs from the myriad to the myriad of myriads, and then from the myriad of myriads to the myriad of myriads of myriads, etc. However, in the 19th century, verification calculations were carried out that showed that perfect numbers were encountered even less so. So, from 1020 to 1036 there is no perfect number, and if you follow Yamblich, then there should be four.
Most likely, it was the difficulty of finding such numbers that served as the reason for endowing them with mystical properties. Although, based on biblical history, its researchers concluded that the world was created really beautiful and perfect, because the number of days of creation is 6. But man is not ideal, since he was created and lives in the seventh day. However, his task is to strive for excellence.
Interesting facts are as follows:
- 8 people were saved in Noah's Ark after the Flood. Also, seven pairs of clean and unclean animals were saved in it. If we summarize all the survivors in the Noah's Ark, then the number 28 is perfect.
- Man’s hands are a perfect tool. They have 10 fingers, which are endowed with 28 phalanges.
- The moon makes near-Earth revolutions every 28 days.
The Pythagoreans considered the number 6 to be psychogonic. The geometric symbol corresponding to 6 is a hexagram.
When drawing a square, you can draw diagonals in it. Then it will be easy to notice that its vertices are connected by 6 segments. If you do the same with the cube, you get 12 edges and 16 diagonals (12 faces, 4 cubes). In total, we get 28. A similar situation will be with the tetrahedron, whose vertices are connected by 6 edges. The octagon is also involved in the perfect number 28 (20 diagonals plus 8 sides). And the seven-sided pyramid has 7 edges and 7 sides of the base with 14 diagonals. In total, this is the number 28.
Interesting calculations
So, perfect is a number equal to the sum of the divisors:
1 + 2 + 3 + ... + n
All divisors that are less than the number itself are added up.
Every ideal number, except 6, is a partial sum of a series consisting of odd numbers in the third degree: 13 + 33 + 53 + ... n³.
Another amazing property of these numbers is as follows: the sum of the inverse values of the divisors, including the one equal to the number itself, will always be 2. For example, take 28, then 1/1 + 1/2 + 1/4 + 1/7 + 1 / 14 + 1/28 = 2.
As mentioned above, all numbers that can be found using the Euclidean formula will be even. Until now, we do not know the odd ideal numbers. Of course, a great breakthrough has recently been made in the science of mathematics and in the issue of perfect numbers in particular. However, the problem of studying these mathematical concepts remains open. Even assuming the existence of an odd ideal number, it would have to be more than 10,300 and have a minimum of 75 prime divisors, given the multiplicity (9 of them should be different).
It is also completely incomprehensible whether the number of perfect numbers is finite, or is it still limited?
All even perfect numbers are equivalent to the sum of consecutive natural numbers. In other words, they are triangular.
Numbers that can be written as 2p - 1 are called Mersenne numbers. Each such number has a corresponding perfect number. The same can be said on the contrary: to each ideal number there corresponds a Mersenne number.
Another important discovery was the connection between duality and perfection. If you look closely, we will see a connection with geometric progression.
Next to the perfect, it is worth noting the friendly numbers. These are two numbers that have a rule: each is equivalent to the sum of the divisors of the second. The smaller ones are 220 and 284. They were familiar to the Pythagoreans. They were given the status of a symbol of friendship. The next couple was opened in 1636. These are 17,296 and 18,416. This friendly couple became known to us thanks to the French lawyer and mathematician Pierre Fermat.
But in 1867, the mathematical world was shocked by the news from sixteen-year-old Italian Niccolo Paganini (namesake of the famous violinist), who reported a friendly pair of numbers 1184 and 1210. It is the closest to 220 and 284. Surprisingly, all the eminent mathematicians who studied friendly numbers overlooked the couple .