Who gave the name to perfect numbers? Perfect number

Eigendivisor a natural number is any divisor other than the number itself. If a number is equal to the sum of its own divisors, then it is called perfect. So, 6 = 3 + 2 + 1 is the smallest of all perfect numbers (1 does not count), 28 = 14 + 7 + 4 + 2 + 1 is another such number.

Perfect numbers have been known since ancient times and have interested scientists at all times. In Euclid's Elements it was proven that if a prime number has the form 2 n– 1 (such numbers are called Mersenne prime numbers), then the number 2 n–1 (2 n– 1) - perfect. And in the 18th century, Leonhard Euler proved that any even perfect number has this form.

Task

Try to prove these facts and find a couple more perfect numbers.

Hint 1

a) To prove a statement from the Principia (what if a prime number has the form 2 n– 1, then the number is 2 n –1 (2n– 1) - perfect), it is convenient to consider the sigma function, which is equal to the sum of all positive divisors of a natural number n. For example, σ (3) = 1 + 3 = 4, and σ (4) = 1 + 2 + 4 = 7. This function has a useful property: it multiplicative, that is σ (ab) = σ (a)σ (b); the equality holds for any two coprime natural numbers a And b (mutually prime are numbers that have no common divisors). You can try to prove this property or take it on faith.

Using the sigma function to prove the perfection of a number N = 2n –1 (2n– 1) comes down to checking that σ (N) = 2N. For this purpose, the multiplicativity of this function is useful.

b) Another solution does not use any additional constructions like the sigma function. It relies only on the definition of a perfect number: you need to write down all the divisors of the number 2 n–1 (2 n– 1) and find their sum. It should be the same number.

Hint 2

Proving that any even perfect number is a power of two multiplied by a Mersenne prime is also convenient using the sigma function. Let N- any even perfect number. Then σ (N) = 2N. Let's imagine N as N = 2k· m, Where m- odd number. That's why σ (N) = σ (2k· m) = σ (2k)σ (m) = (1 + 2 + ... + 2k)σ (m) = (2k +1 – 1)σ (m).

It turns out that 2 2 k· m = (2k +1 – 1)σ (m). So 2 k+1 – 1 divides the product 2 k+1 · m, and since 2 k+1 – 1 and 2 k+1 are relatively prime, then m must be divisible by 2 k+1 – 1. That is m can be written in the form m = (2k+1 – 1) M. Substituting this expression into the previous equality and reducing by 2 k+1 – 1, we get 2 k+1 · M = σ (m). Now there is only one, although not the most obvious, step left until the end of the proof.

Solution

The clues contain much of the evidence for both facts. Let's fill in the missing steps here.

1. Euclid's theorem.

a) First you need to prove that the sigma function is indeed multiplicative. In fact, since every natural number can be uniquely factored into prime factors (this statement is called the fundamental theorem of arithmetic), it is enough to prove that σ (pq) = σ (p)σ (q), Where p And q- various prime numbers. But it is quite obvious that in this case σ (p) = 1 + p, σ (q) = 1 + q, A σ (pq) = 1 + p + q + pq = (1 + p)(1 + q).

Now let's complete the proof of the first fact: if a prime number has the form 2 n– 1, then the number N = 2n –1 (2n– 1) - perfect. To do this, it is enough to check that σ (N) = 2N(since the sigma function is the sum everyone divisors of the number, that is, the sum own divisors plus the number itself). We check: σ (N) = σ (2n –1 (2n – 1)) = σ (2n –1)σ (2n – 1) = (1 + 2 + ... + 2n–1)·((2 n – 1) + 1) = (2n- 12 n = 2N. Here it was used that times 2 n– 1 is a prime number, then σ (2n – 1) = (2n – 1) + 1 = 2n.

b) Let’s complete the second solution. Find all proper divisors of the number 2 n –1 (2n- 1). This is 1; powers of two 2, 2 2, ..., 2 n-1 ; Prime number p = 2n- 1; as well as divisors of type 2 m· p, where 1 ≤ m ≤ n– 2. The summation of all divisors is thereby divided into the calculation of the sums of two geometric progressions. The first one starts with 1, and the second one starts with a number p; both have a denominator equal to 2. According to the formula for the sum of elements of a geometric progression, the sum of all elements of the first progression is equal to 1 + 2 + ... + 2 n –1 = (2n – 1)/2 – 1 = 2n– 1 (and this is equal p). The second progression gives p·(2 n –1 – 1)/(2 – 1) = p·(2 n-eleven). In total, it turns out p + p·(2 n –1 – 1) = 2n-1 · p- what you need.

Most likely, Euclid was not familiar with the sigma function (and indeed with the concept of a function), so his proof is presented in a slightly different language and is closer to the solution from point b). It is contained in sentence 36 of Book IX of Elements and is available, for example, .

2. Euler's theorem.

Before proving Euler's theorem, we also note that if 2 n– 1 is a prime Mersenne number, then n must also be a prime number. The point is that if n = km- compound, then 2 km – 1 = (2k)m– 1 is divisible by 2 k– 1 (since the expression x m– 1 is divided by x– 1, this is one of the abbreviated multiplication formulas). And this contradicts the simplicity of the number 2 n– 1. Converse statement - “if n- prime, then 2 n– 1 is also prime” - not true: 2 11 – 1 = 23·89.

Let's return to Euler's theorem. Our goal is to prove that any even perfect number has the form obtained by Euclid. Hint 2 outlined the first steps of the proof, leaving the final step to take. From equality 2 k+1 · M = σ (m) follows that m divided by M. But m is also divisible by itself. Wherein M + m = M + (2k+1 – 1) M = 2 k+1 · M = σ (m). This means that the number m there are no other divisors except M And m. Means, M= 1, a m- a prime number that has the form 2 k+1 – 1. Then N = 2k· m = 2k(2k+1 – 1), which is what was required.

So, the formulas are proven. Let's use them to find some perfect numbers. At n= 2 the formula gives 6, and when n= 3 turns out to be 28; These are the first two perfect numbers. According to the property of Mersenne prime numbers, we need to choose such a prime n that 2 n– 1 will also be a prime number, and composite n may not be considered at all. At n= 5 equals 2 n– 1 = 32 – 1 = 31, this suits us. Here is the third perfect number - 16·31 = 496. Just in case, let's check its perfection explicitly. Let's write down all the proper divisors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248. Their sum is 496, so everything is in order. The next perfect number is obtained by n= 7 is 8128. The corresponding Mersenne prime is 2 7 – 1 = 127, and it is quite easy to verify that it is indeed prime. But the fifth perfect number is obtained when n= 13 and equals 33,550,336. But checking it manually is already very tedious (however, this did not stop someone from discovering it back in the 15th century!).

Afterword

The first two perfect numbers - 6 and 28 - have been known since time immemorial. Euclid (and we, following him), using the formula we had proven from the Elements, found the third and fourth perfect numbers - 496 and 8128. That is, at first only two were known, and then four numbers with the beautiful property of “being equal to the sum of their divisors " They could not find any more such numbers, and even these, at first glance, had nothing in common. In ancient times, people were inclined to attach mystical meaning to mysterious and incomprehensible phenomena, which is why perfect numbers received a special status. The Pythagoreans, who had a strong influence on the development of science and culture of that time, also contributed to this. “Everything is a number,” they said; The number 6 in their teaching had special magical properties. And the early interpreters of the Bible explained that the world was created precisely on the sixth day, because the number 6 is the most perfect among numbers, for it is the first among them. It also seemed to many that it was no coincidence that the Moon revolves around the Earth in about 28 days.

The fifth perfect number - 33,550,336 - was found only in the 15th century. Almost a century and a half later, the Italian Cataldi found the sixth and seventh perfect numbers: 8,589,869,056 and 137,438,691,328. They correspond to n= 17 and n= 19 in Euclid's formula. Please note that the count is already in the billions, and it’s scary to even imagine that all the calculations were done without calculators and computers!

As we know, Leonhard Euler proved that any even perfect number must have the form 2 n –1 (2n– 1), and 2 n– 1 should be simple. The eighth number - 2 305 843 008 139 952 128 - was also found by Euler in 1772. Here n= 31. After his achievements, one could cautiously say that something became clear to science about even perfect numbers. Yes, they grow quickly and are difficult to calculate, but at least it is clear how to do it: you need to take Mersenne numbers 2 n– 1 and look for simple ones among them. Almost nothing is known about odd perfect numbers. To date, not a single such number has been found, despite the fact that all numbers up to 10,300 have been tested (apparently, the lower limit has been pushed even further, the corresponding results have simply not yet been published). For comparison: the number of atoms in the visible part of the Universe is estimated to be about 10 80. It has not been proven that odd perfect numbers do not exist, it just can be a very large number. Even so large that our computing power will never reach it. Whether such a number exists or not is one of the open problems in mathematics today. The computer search for odd perfect numbers is carried out by participants in the OddPerfect.org project.

Let's return to even perfect numbers. The ninth number was found in 1883 by a rural priest from the Perm province I.M. Pervushin. This number has 37 digits. Thus, by the beginning of the 20th century, only 9 perfect numbers had been found. At this time, mechanical arithmetic machines appeared, and in the middle of the century the first computers appeared. With their help, things went faster. Currently, 47 perfect numbers have been found. Moreover, only the first forty have serial numbers known. About seven more numbers it has not yet been established exactly what they are. The search for new Mersenne primes (and with them new perfect numbers) is mainly carried out by members of the GIMPS project (mersenne.org).

In 2008, project participants found the first prime number with more than 10,000,000 = 10 7 digits. For this they received a prize of $100,000. Cash prizes of $150,000 and $250,000 are also promised for prime numbers consisting of more than 10 8 and 10 9 digits, respectively. It is expected that those who have found smaller but not yet discovered Mersenne primes will also receive a reward from this money. True, on modern computers checking numbers of this length for primality will take years, and this is probably a matter of the future. The largest prime number today is 243112609 – 1. It consists of 12,978,189 digits. Note that thanks to the Lucas-Lehmer test (see its proof: A proof of the Lucas–Lehmer Test), checking for the primality of Mersenne numbers is greatly simplified: there is no need to try to find at least one divisor of the next candidate (this is a very labor-intensive job, which for such large numbers is practically impossible now).

Perfect numbers have some fun arithmetic properties:

• Every even perfect number is also a triangular number, that is, it can be represented as 1 + 2 + ... + k = k(k+ 1)/2 for some k.
• Every even perfect number except 6 is the sum of the cubes of successive odd natural numbers. For example, 28 = 1 3 + 3 3, and 496 = 1 3 + 3 3 + 5 3 + 7 3.
• In the binary number system, the perfect number is 2 n –1 (2n– 1) is written very simply: first they go n units, and then - n– 1 zeros (this follows from Euclid’s formula). For example, 6 10 = 110 2, 28 10 = 11100 2, 33550336 10 = 1111111111111000000000000 2.
• The sum of the reciprocals of all divisors of a perfect number (the number itself is also involved here) is equal to 2. For example, 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2.

Amazing numbers

4.2 Perfect numbers

Sometimes perfect numbers are considered a special case of friendly numbers: every perfect number is friendly to itself. Nicomachus of Geras, the famous philosopher and mathematician, wrote: “Perfect numbers are beautiful. But it is known that things are rare and few in number, ugly things are found in abundance. Almost all numbers are redundant and insufficient, while there are few perfect numbers.” But how many of them are there? Nicomachus, who lived in the first century AD, did not know.

A perfect number is a number equal to the sum of all its divisors (including 1, but excluding the number itself).

The first beautiful perfect number that the mathematicians of Ancient Greece knew about was the number "6". In sixth place at the invited feast lay the most respected, most honored guest. Biblical legends claim that the world was created in six days, because there is no more perfect number among perfect numbers than “6”, since it is the first among them.

Let's consider the number 6. The number has divisors 1, 2, 3 and the number 6 itself. If we add up the divisors other than the number itself 1 + 2 + 3, then we get 6. This means that the number 6 is friendly to itself and is the first perfect number.

The next perfect number known to the ancients was "28". Martin Gardner saw a special meaning in this number. In his opinion, the Moon is renewed in 28 days, because the number “28” is perfect. In Rome in 1917, during underground work, a strange structure was discovered: twenty-eight cells were located around a large central hall. This was the building of the Neopythagorean Academy of Sciences. It had twenty-eight members. Until recently, many learned societies were supposed to have the same number of members, often simply by custom, the reasons for which have long been forgotten. Before Euclid, only these two perfect numbers were known, and no one knew whether other perfect numbers existed or how many such numbers there could be.

Thanks to his formula, Euclid was able to find two more perfect numbers: 496 and 8128.

For almost fifteen hundred years people knew only four perfect numbers, and no one knew whether there could be other numbers that could be represented in the Euclidean formula, and no one could say whether perfect numbers were possible that did not satisfy the Euclid formula.

Euclid's formula allows you to easily prove numerous properties of perfect numbers.

All perfect numbers are triangular. This means that, taking a perfect number of balls, we can always form an equilateral triangle from them.

All perfect numbers except 6 can be represented as partial sums of a series of cubes of successive odd numbers 1 3 + 3 3 + 5 3 ...

The sum of the reciprocals of all divisors of a perfect number, including itself, is always equal to 2.

In addition, the perfection of numbers is closely related to binary. Numbers: 4=22, 8=2? 2? 2, 16 = 2? 2? 2? 2, etc. are called powers of 2 and can be represented as 2n, where n is the number of twos multiplied. All powers of the number 2 fall just a little short of becoming perfect, since the sum of their divisors is always one less than the number itself.

All perfect numbers (except 6) end in decimal notation with 16, 28, 36, 56, 76 or 96.

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Lev Nikolaevich Tolstoy jokingly “bragged that the date of his birth (August 28 according to the calendar of that time) was a perfect number. The year of birth of L.N. Tolstoy (1828) is also an interesting number: the last two digits (28) form a perfect number; and if you rearrange the first two digits, you get 8128 - the fourth perfect number.

Perfect numbers are beautiful. But it is known that beautiful things are rare and few in number. Almost all numbers are redundant and insufficient, but few are perfect.

“What is called perfect is that which, due to its merits and value, cannot be passed in its field” (Aristotle).

Perfect numbers are exceptional numbers; it is not for nothing that the ancient Greeks saw in them some kind of perfect harmony. For example, the number 5 cannot be a perfect number also because the number five forms a pyramid, an imperfect figure in which the base is not symmetrical to the sides.

But only the first two numbers, 6 and 28, were truly deified. There are many examples: in Ancient Greece, the most respected, most famous and honored guest reclined in the 6th place at a banquet; in Ancient Babylon, the circle was divided into 6 parts. The Bible states that the world was created in 6 days, because there is no number more perfect than six. Firstly, 6 is the smallest, the very first perfect number. No wonder the great Pythagoras and Euclid, Fermat and Euler paid attention to him. Secondly, 6 is the only natural number equal to the product of its regular natural divisors: 6=1*2*3. Thirdly, 6 is the only perfect digit. Fourthly, a number consisting of 3 sixes has amazing properties, 666 - the number of the devil: 666 is equal to the sum of the sum of squares of the first seven prime numbers and the sum of the first 36 natural numbers:

666=22+32+52+72+112+132+172,

666=1+2+3++34+35+36.

One interesting geometric interpretation of 6 is that it is a regular hexagon. The side of a regular hexagon is equal to the radius of the circle circumscribed around it. A regular hexagon consists of six triangles with all sides and angles equal. A regular hexagon is found in nature, it is the honeycomb of bees, and honey is one of the most useful products in the world.

Now about 28. The ancient Romans greatly respected this number; in the Roman academies of sciences there were strictly 28 members, in the Egyptian measure the length of a cubit is 28 fingers, in the lunar calendar there are 28 days. But there is nothing about the other perfect numbers. Why? Mystery. Perfect numbers are generally mysterious. Many of their mysteries still cannot be solved, although they thought about it more than two thousand years ago.

One of these mysteries is why the mixture of the most perfect number 6 and the divine 3, the number 666, is the number of the devil. In general, there is something incomprehensible between perfect numbers and the Christian Church. After all, if a person found at least one perfect number, all his sins were forgiven, and life in paradise after death was forgiven. Maybe the church knows something about these numbers that no one would ever think of.

The insoluble mystery of perfect numbers, the powerlessness of the mind before their mystery, their incomprehensibility led to recognition of the divinity of these amazing numbers. One of the most outstanding scientists of the Middle Ages, friend and teacher of Charlemagne, Abbot Alcuin, one of the most prominent figures of education, organizer of schools and author of textbooks on arithmetic, was firmly convinced that the human race is imperfect only for this reason, only for this reason evil and grief reign in it and violence, that he came from eight people who were saved in Noah’s ark from the flood, and “eight” is an imperfect number. The human race before the flood was more perfect - it originated from one Adam, and one can be considered a perfect number: it is equal to itself - its only divisor.

After Pythagoras, many tried to find the following numbers or a formula for their derivation, but only Euclid succeeded in this several centuries after Pythagoras. He proved that if a number can be represented as 2 p-1(2 p-1), and (2 p-1) is prime, then it is perfect. Indeed, if p=2, then 2 2-1(2 2 -1)=6, and if p=3, 2 3-1(2 3 -1)=28.

Thanks to this formula, Euclid found two more perfect numbers, with p=5: 2 5-1(2 5 -1)= 496, 496=1+2+4+8+16+31+62+124+248, and with p= 7: 2 7-1(2 7 -1)=8128, 8128=1+2+4+8+16+32+64+127+254+508+1016+2032+4064.

And again, for almost one and a half thousand years there were no glimmers in the horizon of hidden perfect numbers, until in the 15th century the fifth number was discovered; it also obeyed Euclid’s rule, only with p = 13: 2 13-1 (2 13 -1) = 33550336. Taking a closer look at Euclid's formula, we will see the connection between perfect numbers and the terms of the geometric progression 1, 2, 4, 8, 16; this connection can best be traced using the example of an ancient legend, according to which Raja promised the inventor of chess any reward. The inventor asked to place one grain of wheat on the first square of the chessboard, two grains on the second square, four on the third, eight on the fourth, and so on. The last, 64th cell should contain 264-1 grains of wheat. This is more than has been collected in all harvests in human history. Euclid's formula allows you to easily prove numerous properties of perfect numbers. For example, all perfect numbers are triangular. This means that, taking the perfect number of balls, we can always form an equilateral triangle from them. From the same formula of Euclid follows another curious property of perfect numbers: all perfect numbers, except 6, can be represented as partial sums of a series of cubes of consecutive odd numbers 13+33+53+ Even more surprising is that the sum of the reciprocals of all divisors of a perfect number, including himself, is always equal to 2. For example, taking the divisors of the perfect number 28, we get:

In addition, the representation of perfect numbers in binary form, the alternation of the last digits of perfect numbers and other interesting questions that can be found in the literature on entertaining mathematics are interesting.

Another two hundred years later, the French mathematician Marine Mersenne stated without any evidence that the next six perfect numbers must also be in Euclidean form with p-values ​​of 17, 19, 31, 67, 127, 257. Obviously, Mersenne himself could not verify direct calculation of his statement, because for this he had to prove that the numbers 2 p-1 (2 p -1) with the p values ​​​​he indicated are simple, but then this was beyond human power. So it is still unknown how Mersenne reasoned when he declared that his numbers correspond to the perfect numbers of Euclid. There is an assumption: if you look at the formula for the sum of the first k terms of the geometric progression 1+2+22++2k-2+2k-1, you can see that the Mersenne numbers are nothing more than simple sums of the terms of the geometric progression with base 2:

67=1+2+64, etc.

A generalized Mersenne number can be called the simple value of the sum of the terms of a geometric progression with base a:

1+a+a2++ak-1=(ak-1)/a-1.

It is clear that the set of all generalized Mersenne numbers coincides with the set of all odd prime numbers, since if k is prime or k>2, then k=(k-2)k/k-2=(k-1)2-1/( k-1)-1.

Now everyone can independently explore and calculate Mersenne numbers. Here is the beginning of the table.

and k- for which ak-1/a-1 are simple

Currently, Mersenne primes are used to protect electronic information and are also used in cryptography and other applications of mathematics.

But this is only an assumption; Mersenne took his secret with him to the grave.

The next in a series of discoveries was the great Leonhard Euler, he proved that all even perfect numbers have the form indicated by Euclid and that the Mersenne numbers 17, 19, 31 and 127 are correct, but 67 and 257 are not correct.

Р=17.8589869156 (sixth number)

Р=19.137438691328 (seventh number)

P=31.2305843008139952128 (eighth number).

The ninth number was found in 1883, having accomplished a real feat, because he counted without any instruments, by a rural priest from near Perm, Ivan Mikheevich Pervushin, he proved that 2p-1, with p = 61:

2305843009213693951 is a prime number, 261-1(261-1)= 2305843009213693951*260 – it has absolutely 37 digits.

At the beginning of the 20th century, the first mechanical calculating machines appeared, which ended the era when people counted by hand. With the help of these mechanisms and computers, all other perfect numbers that are now known were found.

The tenth number was discovered in 1911 and has 54 digits:

618970019642690137449562111*288, p=89.

The eleventh, with 65 digits, was discovered in 1914:

162259276829213363391578010288127*2106, p=107.

The twelfth was also found in 1914, 77 digits p=127:2126(2127-1).

The fourteenth was discovered on the same day, 366 digits p=607, 2606(2607-1).

In June 1952, the 15th number 770 digits p = 1279, 21278 (21279-1) was found.

The sixteenth and seventeenth opened in October 1952:

22202(22203-1), 1327 digits p=2203 (16th number)

22280(22281-1), 1373 digits p=2281 (17th number).

The eighteenth number was found in September 1957, 2000 digits p = 3217.

The search for subsequent perfect numbers required more and more calculations, but computer technology was constantly improving, and in 1962 2 numbers were found (p = 4253 and p = 4423), in 1965 three more numbers (p = 9689, p = 9941, p =11213).

More than 30 perfect numbers are now known, the largest p is 216091.

But this, in comparison with the riddles that Euclid left: whether there are odd perfect numbers, whether the series of even Euclidean perfect numbers is finite, and whether there are even perfect numbers that do not obey Euclid’s formula - these are the three most important riddles of perfect numbers. One of which was solved by Euler, who proved that there are no even perfect numbers other than Euclidean ones. 2 The rest remain unsolved even in the 21st century, when computers have reached such a level that they can perform millions of operations per second. The existence of an odd imperfect number and the existence of a greatest perfect number are still not resolved.

Without a doubt, perfect numbers live up to their name.

Among all the interesting natural numbers that have long been studied by mathematicians, perfect numbers and closely related friendly numbers occupy a special place. These are two numbers, each of which is equal to the sum of the divisors of the second friendly number. The smallest friendly numbers, 220 and 284, were known to the Pythagoreans, who considered them a symbol of friendship. The next pairs of friendly numbers 17296 and 18416 were discovered by the French lawyer and mathematician Pierre Fermat only in 1636, and subsequent numbers were found by Descartes, Euler and Legendre. 16-year-old Italian Niccolo Paganini (namesake of the famous violinist) shocked the mathematical world in 1867 with the message that the numbers 1184 and 1210 are friendly! This pair, closest to 220 and 284, was overlooked by all the famous mathematicians who studied friendly numbers.

And at the end it is proposed to solve the following problems related to perfect numbers:

1. Prove that a number of the form 2 р-1(2 р -1), where 2к-1 is a prime number, is perfect.

2. Let us denote by, where is a natural number, the sum of all its divisors. Prove that if the numbers are relatively prime, then.

3. Find more examples that perfect numbers were very revered by the ancients.

4. Look carefully at a fragment of Raphael’s painting “The Sistine Madonna.” What does it have to do with perfect numbers?

5. Calculate the first 15 Mersenne numbers. Which of them are prime and which perfect numbers correspond to them.

6. Using the definition of a perfect number, imagine one as the sum of different unit fractions whose denominators are all the divisors of the given number.

7. Arrange 24 people in 6 rows so that each row contains 5 people.

8. Using five twos and arithmetic spells, write down the number 28.

The number 6 is divisible by itself, and also by 1, 2 and 3, and 6 = 1+2+3.
The number 28 has five factors other than itself: 1, 2, 4, 7 and 14, with 28 = 1+2+4+7+14.
It can be noted that not every natural number is equal to the sum of all its divisors that differ from this number. Numbers that have this property have been named perfect.

Even Euclid (3rd century BC) indicated that even perfect numbers can be obtained from the formula: 2 p –1 (2p– 1) provided that R and 2 p There are prime numbers. In this way, about 20 even perfect numbers were found. Until now, not a single odd perfect number is known and the question of their existence remains open. Research into such numbers was begun by the Pythagoreans, who attributed a special mystical meaning to them and their combinations.

The first smallest perfect number is 6 (1 + 2 + 3 = 6).
Perhaps that is why the sixth place was considered the most honorable at feasts among the ancient Romans.

The second highest perfect number is 28 (1 + 2 + 4 + 7 + 14 = 28).
Some learned societies and academies were supposed to have 28 members. In Rome in 1917, while carrying out underground work, the premises of one of the oldest academies were discovered: a hall and around it 28 rooms - just the number of members of the academy.

As natural numbers increase, perfect numbers become less common. Third perfect number - 496 (1+2+48+16+31+62+124+248 = 496), fourth – 8128 , fifth - 33 550 336 , sixth - 8 589 869 056 , seventh - 137 438 691 328 .

The first four perfect numbers are: 6, 28, 496, 8128 were discovered a long time ago, 2000 years ago. These numbers are given in the Arithmetic of Nicomachus of Geraz, an ancient Greek philosopher, mathematician and music theorist.
The fifth perfect number was discovered in 1460, about 550 years ago. This number 33550336 discovered by the German mathematician Regiomontanus (15th century).

In the 16th century, the German scientist Scheibel also found two more perfect numbers: 8 589 869 056 And 137 438 691 328 . They correspond to p = 17 and p = 19. At the beginning of the 20th century, three more perfect numbers were found (for p = 89, 107 and 127). Subsequently, the search slowed down until the middle of the 20th century, when, with the advent of computers, calculations beyond human capabilities became possible. There are currently 47 even perfect numbers known.

The perfect nature of the numbers 6 and 28 was recognized by many cultures, noting that the Moon orbits the Earth every 28 days, and claiming that God created the world in 6 days.
In his essay “The City of God,” St. Augustine expressed the idea that although God could create the world in an instant, He chose to create it in 6 days in order to reflect on the perfection of the world. According to St. Augustine, the number 6 is absolutely not because God chose it, but because perfection is inherent in the nature of this number. “The number 6 is perfect in itself, and not because the Lord created all things in 6 days; rather, on the contrary, God created everything that exists in 6 days because this number is perfect. And it would remain perfect even if there was no creation in 6 days.”

Lev Nikolaevich Tolstoy more than once jokingly “boasted” that the date
his birth on August 28 (according to the calendar of that time) is a perfect number.
Year of birth L.N. Tolstoy (1828) is also an interesting number: the last two digits (28) form a perfect number; If you swap the first digits, you get 8128 - the fourth perfect number.

Karatetskaya Maria

In this abstract work with elements of independent research, the concept of a perfect number is “discovered”,

The properties of perfect numbers, the history of their appearance are explored, and interesting facts related to the concept are presented.

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“Secondary school No. 19 with in-depth study

Individual items"

Scientific Society of Students “Clever Men and Clever Girls”

Abstract work with elements

independent research

"Perfect Numbers"

Performed:

7th grade student "A"

Karatetskaya Maria

Supervisor:

mathematic teacher

Kolina Natalya Konstantinovna

OS address:

606523, Nizhny Novgorod region, Gorodetsky

District, Zavolzhye, Molodezhnaya st., 1

MBOU secondary school No. 19 with UIOP

Email: [email protected]

2015

1.Introduction………………………………………………………………………………………3

2.What is a perfect number?……...........................………….......... ..................4

3. The history of the appearance of perfect numbers………………………………………....4

4. Properties of perfect numbers…………………………….………………………....8

5. Interesting facts…………………………………..………………...................8

6. Examples of tasks………………………………………………………………………………….9

7. Conclusion…………………………………………………………….........11

8. List of references used…………………………….………………......12

“Everything is beautiful thanks to number” Pythagoras.

1. Introduction

Number is one of the basic concepts of mathematics. There are a large number of definitions for the concept “number”. Pythagoras was the first to talk about numbers. According to his teachings, the number 2 meant harmony, 5 - color, 6 - cold, 7 - intelligence, health, 8 - love and friendship. The first scientific definition of number was given by Euclid in his work “Elements”: “A unit is that in accordance with which each of the existing things is called one. A number is a set made up of units.”

There are sets of numbers, their subsets, groups, and one of the unusual groups is perfect numbers. Only 48 numbers are known in this group, but despite this, theyform one of the most interesting subsets of the set of natural numbers.

Problem: I love solving non-standard problems. One day I came across a problem that talked about perfect numbers, I had difficulty solving it, so I became interested in this topic and decided to study these numbers in more detail.

Purpose of the study:get acquainted with the concept of a perfect number, explore the properties of perfect numbers,attract students' attention to this topic.

Tasks:

Study and analyze the literature on the research topic.

Study the history of the appearance of perfect numbers.

-“Discover” the properties of perfect numbers and their areas of application

Expand your mental horizons.

Research methods:literature study, comparison, observation,

theoretical analysis, generalization.

2.What is a perfect number?

Perfect number- natural number , equal to the sum of all itsproper divisors (i.e. all positive divisors, including 1, but different from the number itself).

First perfect numberhas the following proper divisors: 1, 2, 3; their sum 1 + 2 + 3 is 6.

Second perfect numberhas the following proper divisors: 1, 2, 4, 7, 14; their sum 1 + 2 + 4 + 7 + 14 is 28.

The third perfect number 496 has the following proper divisors: 1, 2, 4, 8, 16, 31, 62, 124, 248; their sum 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 is 496.

The fourth perfect number ishas the following proper divisors: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064; their sum 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 is 8128.

As natural numbers increase, perfect numbers become less common.

3. The history of the appearance of perfect numbers

Ancient Greek mathematician and philosopher Pythagoras , he is also the creator of the religious and philosophical school of the Pythagoreans (570-490 BC), introduced the concepts of excess and insufficient numbers.

If the sum of the divisors of a number is greater than the number itself, then such a number is called “excess”. For example, 12 is a redundant number because the sum of its divisors is 16. If the sum of the divisors of a number is less than the number itself, then such a number is called “insufficient.”

For example, 10 is not a sufficient number because the sum of its divisors (1, 2 and 5) is only 8.

The Pythagoreans developed their philosophy from the science of numbers. Perfect numbers, they believed, are beautiful images of virtues. They represent the middle ground between excess and deficiency. They are very rare and are generated by perfect order. In contrast to this, superabundant and imperfect numbers, of which there are as many as possible, are not arranged in order and are not generated for some specific purpose. And therefore they have a great resemblance to vices, which are numerous, not ordered and not defined.

“A perfect number is equal to its shares.” These words belong Euclid , ancient Greek mathematician, author of the first theoretical treatise on mathematics that has come down to us, “Elements” (3rd century BC).Before Euclid, only two perfect numbers were known, and no one knew whether there were other perfect numbers or how many such numbers there could be. Thanks to its formula 2 p-1 *(2 p -1) is a perfect number if (2 p -1) is a prime number. So Euclid managed to find two more perfect numbers: 496 and 8128. The method of finding perfect numbers is described in Book IX of the Elements.

Nicomachus of Geraz, Greek philosopher and mathematician (1st half of the 2nd century AD), in his essay “Introduction to Arithmetic” wrote: “...Beautiful and noble things are usually rare and easily counted, while ugly and bad things are numerous; So, excess and insufficient numbers are found in large numbers and randomly, so that the method of finding them is not ordered, while perfect numbers are easily enumerable and arranged in the proper order. Indeed, among single-digit numbers there is one such number 6, the second number 28 is the only one among tens, the third number 496 is the only one among hundreds, and the fourth number 8128 is among thousands, if we limit ourselves to ten thousand. And their inherent property is that they alternately end in a six, then in an eight, and are all even. An elegant and reliable way of obtaining them, which does not miss a single perfect number and gives only perfect numbers, is as follows. Arrange all even-even numbers, starting with one, in one row, continuing it as far as you wish: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.

Then add them sequentially, adding one each time,

and after each addition look at the result; and when will he be

primary and non-compound, multiply it by the last added

number, resulting in you always getting a perfect number.

If it is secondary and composite, there is no need to multiply, but it is necessary

add the next number and look at the result; if he again

turns out to be secondary and composite, skip it again and do not multiply, but

add the following; but if it is primary and non-composite, then

multiplying it by the last added number, you will again get

perfect number, and so on ad infinitum. And in this way you

you will get all the perfect numbers in order without missing a single one

of them. For example, I add 2 to 1 and see what number I get

in total, and I find that this number is 3, primary and non-composite in agreement

with what was said above, since it does not have different names

share with him, but only the share named after him; now I multiply

it to the last added number, which is 2, and I get 6; and I

I declare it to be the first real perfect number having

such shares that, when put together, they fit into

number itself: after all, the unit is named after it, oh there is

the sixth, a beat, and 3 is a half in accordance with the number 2, and

back, two is a third. The number 28 is obtained in the same way, when the next number 4 is added to those already added

higher. After all, three numbers 1, 2, 4 add up to the number 7, which turns out to be

primary and non-composite, since it has only a name

to him a seventh share; and therefore I multiply it by the last amount,

added to the sum, and my result is 28, equal to my

shares, and having shares named after the numbers already mentioned:

half for fourteen, quarter for seven, seventh for

4, fourteenth as opposed to half, twenty-eighth

in accordance with its own name, and such a fraction for all numbers is equal to one. And when they are already open in units of 6 and in tens of 28, you

8, and you get 15; looking at it, I find out that it is not

primary and non-composite, because in addition to the one named after it

share it has shares opposite to it, the fifth and third; that's why I don't

I multiply it by 8, but add the next number 16 and get the number

31. It is primary and incomposite, and therefore it is necessary, in

as a general rule, multiply by the last added number 16, resulting in 496 in hundreds; and then it becomes 8128 in thousands; and so on, as long as there is a desire to continue..."

It should be said that by secondary number Nicomachus understands a number that is a multiple of a given one, that is, one that can be obtained by multiplying by natural numbers; He refers to the factors included in the expansion of a number as fractions.

If Nicomachus of Geraz found only the first 4 perfect numbers, then Regiomontan( real name - Johann Muller), a German mathematician who lived in the 15th century, found the fifth perfect number - 33550336.

In the 16th century, a German scientistJohann Ephraim Scheibelfound two more perfect numbers - 8589869056 (8 billion, 589 million, 869 thousand, 56), 137438691328 (137 billion, 438 million, 691 thousand, 328).

Cataldi Pietro Antonio(1548-1626), who was a professor of mathematics in Florence and Bologna, who was the first to give a method for extracting square roots, also searched for perfect numbers. His notes indicated the meanings of the sixth and seventh perfect numbers. 8 589 869 056 (sixth number), 137 438 691 328 (seventh number) for p=17 and 19)

17th century French mathematician Maren Mersenne predicted that many numbers described by the formula, where p is a prime number, are also prime. He managed to prove that for p=17, p=19, p=31 the numbers 8589869056, 137438691328, 2305843008139952128 are perfect.

Swiss, German and Russian mathematician and mechanic, who made a fundamental contribution to the development of these sciences, Leonard Euler (early 18th century) proved that all even perfect numbers correspond to the algorithm for constructing even perfect numbers, which is described in Book IX of Euclid’s Elements. He also proved that every even perfect number has the formMp, where the Mersenne number Mp is prime.

The ninth perfect number was not calculated until 1883. It contained thirty-seven characters. This computational feat was performed by a rural priest from near Perm.Ivan Mikheevich Pervushin. Pervushin counted without any computing devices.

At the beginning of the 20th century, three more perfect numbers were found (for p = 89, 107 and 127).

As of February 2013, 48 Mersenne primes and their corresponding even perfect numbers are known; distributed computing projects GIMPS and OddPerfect.org are searching for new Mersenne primes.

4. Properties of perfect numbers

1. All even perfect numbers (except 6) are the sum of cubes of consecutive odd natural numbers.

2. All even perfect numbers are triangular numbers; in addition, they are hexagonal numbers, that is, they can be represented in the form n(2n−1) for some natural number n.

3. The sum of all numbers inverse to the divisors of a perfect number (including itself) is equal to 2, that is

4. All even perfect numbers, except 6 and 496, end in decimal notation with 16, 28, 36, 56 or 76.

5.All even perfect numbers in binary notation contain first p units followed by p -1 zeros (a consequence of their general representation).

6. It has been proven that an odd perfect number, if it exists, has at least 9 different prime factors and at least 75 prime factors, taking into account multiplicity.

5. Interesting facts

Due to the difficulty of finding and mysterious incomprehensibility, perfect numbers were considered divine in ancient times. Thus, the medieval church believed that the study of perfect numbers leads to the salvation of the soul, and that those who find a new perfect number are guaranteed eternal bliss. In the 12th century, the church argued that to save the soul it was necessary to find the fifth perfect number. There was also a belief that the world was beautiful because it was created by the creator in 6 days. But the human race, they say, is imperfect, because it originated from the imperfect number 8. After all, it was 8 people who were saved from the global flood in Noah’s ark. It can be added that in the same ark seven more pairs of clean and seven pairs of unclean animals were saved, which in total makes up the perfect number 28. And in general it is easy to detect many similar coincidences. For example, human hands can be declared a perfect tool for the reason that there are 28 phalanges in ten fingers...

The Egyptian measure of length "cubit" contained 28 fingers.

In sixth place at the banquet sat the most respected, most honored guest.

In 1917, during underground work, a strange structure was discovered: twenty-eight cells were located around a large central hall. Later they learned that this was the building of the Neo-Pythagorean Academy of Sciences. It had twenty-eight members.

Even now, following ancient tradition, some academies according to their charter consist of 28 full members. Despite the fact that perfect numbers have a mystical meaning, Mersenne numbers were completely useless for a long time, just like perfect numbers. But Mersenne primes are now the basis for the security of electronic information, and they are also used in cryptography and other applications of mathematics.

Lev Nikolaevich Tolstoy playfully “bragged” that the date of his birth (August 28 according to the calendar of that time) was a perfect number. The year of birth of Leo Tolstoy (1828) is also an interesting number: the last two digits (28) form a perfect number; and if you rearrange the first two digits, you get 8128 - the fourth perfect number.

6. Examples of problems

1.Find all perfect numbers up to 1000.

Answer: 6 (1+2+3=6), 28 (1+2+4+7+14=28), 496 (1 + 2 + 4 + 8 + 16 + 31 + 62 +

124 + 248 = 496). There are 3 numbers in total.

2.Find a perfect number that is greater than 496 but less than 33550336.

Answer: 8128.

3. A perfect number greater than 6 is divisible by 3. Prove that it is divisible by 9.

Solution: the opposite method. Suppose that a perfect number divisible by 3 is not a multiple of 9. Then it is equal to 3n, where n is not a multiple of 3. Moreover, all natural divisors of 3n (including itself) can be

split into pairs d and 3d, where d is not divisible by 3. Therefore, the sum of all

divisors of the number 3n (it is equal to 6n) is divisible by 4. Hence n is a multiple of 2. Next

note that the numbers 3n/2, n, n/2 and 1 will be different divisors of the number 3n,

their sum is 3n + 1 > 3n, which means that the number 3n cannot be

perfect. Contradiction. This means that our assumption is incorrect and the statement is proven.

4. A perfect number greater than 28 is divisible by 7. Prove that it is divisible by 49.

7.Conclusion

Pythagoras deified numbers. He taught: numbers rule the world. The omnipotence of numbers is manifested in the fact that everything in the world is subject to numerical relations. The Pythagoreans sought in these relationships both the patterns of the real world and the path to mystical secrets and revelations. Numbers, they taught, are characterized by everything - perfection and imperfection, finitude and infinity.

Having considered one of the groups of natural numbers - perfect numbers, I concluded that the variety of natural numbers is infinite. As for the statement that among perfect numbers there are both even and odd numbers, it cannot be considered true, since all perfect numbers discovered so far are even. No one knows whether there is at least one odd perfect number, or that the set of perfect numbers is infinite.

In the future I want to explore friendly numbers.

Friendly numbers are two different natural numbers for which the sum of all proper divisors of the first number is equal to the second number and vice versa, the sum of all proper divisors of the second number is equal to the first number. An example of such a pair of numbers is the pair 220 and 284. Perfect numbers are considered a special case of friendly numbers: every perfect number is friendly to itself. Although these pairs are not of great importance for number theory, they are an interesting element of entertaining mathematics.

8. List of used literature

1. Volina V.V. Entertaining mathematics for children./Ed. V. V. Fedorov; Hood. T. Fedorova. – St. Petersburg: Lev and K°, 1996. – 320 p.
2. Universal school encyclopedia. T. 1. A – L/Chapter. ed. E. Khlebalina, leading ed. D. Volodikhin. – M.: Avanta+, 2003. – 528 p.
3. Universal school encyclopedia. T. 2. A – L/Chapter. ed. E. Khlebalina, leading ed. D. Volodikhin. – M.: Avanta+, 2003. – 528 p.
4. Electronic children's encyclopedia Cyril and Methodius (version 2007).
5. Electronic site WikipediA/ http://www.wikipedia.org/
6. http://eschool.karelia.ru/petrozavodsk/projects/zpivkoren/Lists/List/DispForm.aspx?ID=18
7. http://www.ngpedia.ru/id598396p3.html
8. http://www.ngpedia.ru/id598396p1.html
9. http://academic.ru/dic.nsf/bse/133758/%D0%A1%D0%BE%D0%B2%D0%B5%D1%80%D1%88%D0%B5%D0%BD%D0 %BD%D1%8B%D0%B5
10. http://arbuz.narod.ru/z_sov1.htm