Perfect number. What are perfect numbers in mathematics

§ 2. Mutually prime numbers The number 1 is a common divisor for any pair of numbers a and b. It may happen that unity will be their only common divisor, i.e. d0 = D(a, b) = 1. (4.2.1) In this case, we say that the numbers a and b are relatively prime. Example. (39, 22) = 1.If the numbers have a common

§ 1. Numbers “Everything is a number” - taught the ancient Pythagoreans. However, the number of numbers they used is insignificant compared to the fantastic dance of numbers that surrounds us today in everyday life. Huge numbers appear when we count, and when

44. What numbers? What two integers, if multiplied, make seven? Remember that both numbers must be integers, so answers like 31/2? 2 or 21/3? 3, not

47. Three numbers Which three integers, if multiplied, give the same amount as obtained from them From the author’s book

Magic Numbers As with many previous surveys, respondents found that the average number of lifetime sexual partners was relatively low: about seven for heterosexual women and about thirteen for heterosexual men.

Chapter 1 Numbers Albert! Stop telling God what to do! Niels Bohr to Albert Einstein In the beginning there were number and figure. When man tried to master them, science was born, and man began to learn the world. The development of science was often accompanied by funny,

Appendix Curly Numbers A figurative number is a number that can be represented as points arranged in the shape of a regular polygon. These numbers for a long time served as the object of close attention of mathematicians. The Greeks attributed magical properties to them,

Examples

• 1st perfect number - has the following proper divisors: 1, 2, 3; their sum 1 + 2 + 3 is 6.
• 2nd perfect number - has the following proper divisors: 1, 2, 4, 7, 14; their sum 1 + 2 + 4 + 7 + 14 is 28.
• 3rd perfect number - 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.
• 4th perfect number - has 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.

History of the study

Even perfect numbers

The algorithm for constructing even perfect numbers is described in Book IX Began Euclid, where it was proven that a number is perfect if the number is prime (the so-called Mersenne primes). Subsequently, Leonhard Euler proved that all even perfect numbers have the form indicated by Euclid.

The first four perfect numbers are given in Arithmetic Nicomacheus of Geraz. The fifth perfect number, 33,550,336, was discovered by the German mathematician Regiomontanus (15th century). In the 16th century, the German scientist Scheibel found two more perfect numbers: 8,589,869,056 and 137,438,691,328. They correspond to R= 17 and R= 19. At the beginning of the 20th century, three more perfect numbers were found (for R= 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.

As of April 2010, 47 Mersenne primes and their corresponding even perfect numbers are known; the GIMPS distributed computing project is searching for new Mersenne primes.

Odd perfect numbers

Odd perfect numbers have not yet been discovered, but it has not been proven that they do not exist. It is also unknown whether the set of all perfect numbers is infinite.

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. The distributed computing project OddPerfect.org is looking for odd perfect numbers.

Properties

Notable Facts

The special (“perfect”) nature of the numbers 6 and 28 was recognized in cultures based on the Abrahamic religions, which claim that God created the world in 6 days and noted that the Moon orbits the Earth in approximately 28 days.

“Equally important is the idea expressed by the number 496. It is the “theosophical extension” of the number 31 (that is, the sum of all the integers from 1 to 31). Among other things, it is the sum of the word Malkuth, meaning "Kingdom". Thus the Kingdom, the full manifestation of the primary idea of ​​God, appears in gematria as the natural complement or manifestation of the number 31, which is the number of the name 78.”

"The number 6 is perfect in itself, and not because the Lord created all things in 6 days; rather, on the contrary, God created all things in 6 days because this number is perfect. And it would remain perfect even if there was no creation in 6 days."

see also

• Slightly redundant numbers (quasi-perfect numbers)

Notes

Links

• Depman I. Perfect numbers // Quantum. - 1991. - No. 5. - P. 13-17.

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See what “Perfect number” is in other dictionaries:

PERFECT NUMBER, see NUMBER PERFECT...

A natural number equal to the sum of all its regular (i.e., smaller than this number) divisors. For example, 6=1+2+3 and 28=1+2+4+7+14 are perfect numbers... Big Encyclopedic Dictionary

A natural number equal to the sum of all its regular (that is, smaller than this number) divisors. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are perfect numbers. * * * PERFECT NUMBER PERFECT NUMBER, a natural number equal to the sum... ... encyclopedic Dictionary

Whole positive number, having the property that it coincides with the sum of all its positive divisors other than this number itself. Thus, an integer is a number if the number is, for example, the numbers 6, 28, 496, 8128,33550336... Mathematical Encyclopedia

NUMBER, PERFECT, INTEGER equal to the sum of its DIvisors, including 1. For example, the number 28 is a perfect number because its divisors are the numbers 1, 2, 4, 7 and 14 (not counting the number 28 itself), and their sum is 28 It is not known... ... Scientific and technical encyclopedic dictionary

Numbers of the form Mn = 2n 1, where n is a natural number. Named after the French mathematician Mersenne. The sequence of Mersenne numbers begins like this: 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, ... (sequence A000225 in OEIS) Sometimes numbers ... ... Wikipedia

Number- Since ancient times different numbers attributed secret meanings. Philosophers, followers of Pythagoras (about 500 BC), argued that numbers are the basic principle and essence of things and defined in detail the qualities and types of numbers. According to them... ... Dictionary of Biblical Names

Continuous closed topological mapping. spaces such that the inverse images of all points are compact. S. o. are in many ways similar to continuous mappings of compact spaces into Hausdorff spaces (each such mapping is perfect), but a sphere... ... Mathematical Encyclopedia

Hexagonal number is a curly number. The nth hexagonal number is the number of points in a hexagon with exactly n points on each side. Formula for the nth hexagonal number ... Wikipedia

This term has other meanings, see 6 (meanings). 6 six 3 4 5 6 7 8 9 Factorization: 2×3 Roman notation: VI Binary: 110 Octal: 6 Hex... Wikipedia

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 useful property: she 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 structures like a 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 tips contain Substantial part evidence of 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 very big 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.

When dealing with large numbers, scientists use powers of 10 to get rid of huge amount zeros. For example, 19,160,000,000,000 miles can be written as 1.916 10 13 miles. Very much the same small number, for example 0.0000154324 g, can be written 1.54324·10 –5 g. Of the prefixes used before numerals, the smallest value corresponds to atto, which comes from the Danish or Norwegian atten - eighteen. The prefix means 10 –18. The prefix exa (from the Greek hexa, i.e. 6 groups of 3 zeros), or abbreviated E, means 10 18.

Biggest numbers

The most a large number, found in explanatory dictionaries and called a power of 10, is the centillion, first used in 1852. It is a million to the hundredth power, or one followed by 600 zeros.

The largest named non-decimal number is the Buddhist number asankheya, equal to 10 140; it is mentioned in the Jaina Sutra works dating back to 100 BC.

The number 10,100 is called googol. This term was coined by the 9-year-old nephew of Edward Kasner (USA) (d. 1955). A googol 10 to the power is called a googolplex. Some idea of ​​this value can be obtained by remembering that the number of electrons in the observable Universe, according to some theories, does not exceed 10 87 .

The largest number ever used in mathematical proof is the limiting quantity known as Graham's number, first used in 1977. It is associated with bichromatic hypercubes and cannot be expressed without a special 64-level system of special mathematical symbols introduced by Knuth in 1977

Highest number of multipliers

Computer scientists, using more than 400 interconnected computers, found the factors of a 100-digit number. The calculations, which took 26 days, call into question the reliability of many modern encryption systems.

Prime numbers

A prime number is any positive integer (except 1) that is divisible only by itself or by one, i.e. 2, 3, 5, 7 or 11. The smallest prime number is 2. The largest prime number, 391,581 2 216193 – 1, was discovered on August 6, 1989 by the group Amdal-6. The number containing 65,087 characters was obtained on the Amdal-1200 supercomputer in Santa Clara, California, USA. The team also discovered the largest paired prime numbers: (1,706,595 2 11235 – 1) and (1,706,595 2 11235 + 1). The smallest non-prime or composite number (other than 1) is 4.

Perfect numbers

A number is perfect if it is equal to the sum of its divisors other than the number itself, for example 1 + 2 + 4 + 7 + 14 = 28. The smallest perfect number is: 6 = 1 + 2 + 3.

The largest known number, the 31st discovered to date, is (2 216091 – 1) 2 216090 . This number was obtained thanks to the discovery in September 1985 by the mathematician Marcenne (USA) of the number 2 216091 - 1, which is currently known as the second largest prime number.

Newest mathematical constant

In the course of studies of turbulent water flow, weather and other chaotic phenomena, the existence of a new universal constant was revealed - the Feigenbaum number, named after its discoverer. It is approximately equal to 4.669201609102990.

Maximum number of proofs of the theorem

Longest proof

Proof of the classification of all finite simple groups took up more than 14 thousand pages containing almost 500 scientific works, the authors of which were more than 100 mathematicians. The proof continued for more than 35 years.

The oldest mathematical problem

It dates back to 1650 BC. and in the Russian version it sounds like this:

On the way to Dijon
I met a husband and his seven wives.
Each wife has seven bales,
Each bale contains seven cats.
How many cats, bales and wives
Moved peacefully to Dijon?

The largest number claimed to be accurate in physics

The English astronomer Sir Arthur Eddington (1882...1944) stated in 1938 that there are exactly 15,747,724,136,275,002,577,605,653,961,181,555 468,044 in the Universe in the Universe 076 185 631 031 296 protons and the same number of electrons. Unfortunately for Eddington, no one agreed with his ultra-precise calculations, which are currently not taken seriously.

Most prolific mathematician

Leonhard Euler (Switzerland, Russia) (1707...1783) was so prolific that more than 50 years after his death his works were still being published for the first time. The collected works have been published in parts since 1910, and will ultimately amount to 75 large volumes in quarto size.

The biggest prize

Dr. Paul Wolfskell bequeathed in 1908 a prize of 100 thousand German marks to the first person to prove Fermat's Last Theorem. As a result of inflation, the premium is now just over DM 10,000.

The longest search on a computer for an answer to the question: yes or no?

Fermat's 20th number + 1 was tested on the Cray-2 supercomputer in 1986 to determine whether it was prime. After 10 days of calculations, the answer was NO.

The most mathematically illiterate

The Nambikwara people, living in the northwestern state of Mato Grosso, Brazil, are the most math illiterate. They completely lack a number system. True, they use a verb that means “they are equal.”

The most accurate and inaccurate value of the number π

The largest number of decimal places for π, equal to 1,011,196,691 decimal places, was obtained in 1989 by David and Gregory Chudnovsky of Columbia University, New York, USA, using the Cray-2 supercomputer and a network of IBM 3090 computers. Calculations have been checked for accuracy. By the way, decimal placesπ from the 762nd to the 767th decimal place contain 6 nines in a row.

In 1897, the General Assembly of the American state of Indiana approved Bill 246, according to which the number π was taken to be equal to 4. In 1853, William Shanks published his calculations of the number π to the 707th decimal place, made by hand. 92 years later, in 1945, it was discovered that the last 180 digits were incorrect.

The most ancient units of measurement

The oldest known measure of weight is beka from the Amratian period of Egyptian civilization (circa 3800 BC), found in Naqada, Egypt. The weights were cylindrical in shape with rounded ends. They weighed from 188.7 to 211.2 g.

Apparently, the builders of megalithic tombs in northwestern Europe (about 3500 BC) used a length measure of 82.9 ± 0.09 cm. This conclusion was reached by Professor Alexander Thom (1894... 1985) in 1966

Measuring time

Due to changes in the length of the day, which increase on average by 1 ms per century under the influence of the tidal forces of the Moon, the definition of the second was revised. Instead of 1/86,400 of the average solar day, its duration since 1960 has been defined as 1/315,569,259,747 of the solar (or tropical) year as of 12 hours of ephemeris time in January 1900. In 1958, the second was taken equal to 9,192 631,770 ± 20 periods of radiation corresponding to the transition between the ground state levels of the cesium-133 atom in the absence of external fields. The largest daily change recorded was on August 8, 1972, which was 10 ms and was caused by the most powerful solar storm observed in the last 370 years.

The accuracy of the cesium frequency standard approaches 8 parts in 10 14 , which is higher than 2 parts in 10 13 for a methane-stabilized helium-neon laser and than 6 parts in 10 13 for a hydrogen maser.

The longest measure of time is kalpa in Hindu chronology. It is equal to 4320 million years. In astronomy, a cosmic year is the period of revolution of the Sun around the center Milky Way, it is equal to 225 million years. During the Late Cretaceous period (about 85 million years ago), the Earth rotated faster, resulting in a year consisting of 370.3 days. There is also evidence that in the Cambrian era (600 million years ago) the year lasted more than 425 days.

Guinness Book of Records, 1998