The beauty of numbers. Anti-prime numbers

The number 60 has twelve dividers: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Everyone knows about the amazing properties of prime numbers, which are divided only into themselves and into units. These numbers are extremely useful. Relatively large primes (from about ^10,300 ) are used in public-key cryptography, in hash tables, to generate pseudo-random numbers, etc. In addition to the enormous benefits for human civilization, these particular numbers are strikingly beautiful:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 ...

All other natural numbers greater than one that are not prime are called composite. They have several dividers. So, among the composite numbers there is a special group of numbers that can be called “super-composite” or “anti-simple”, because they have especially many divisors. Such numbers are almost always redundant (except 2 and 4).

Redundancy is a positive integer N, whose sum of its own divisors (except N) exceeds N.
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For example, the number 12 immediately has six dividers: 1, 2, 3, 4, 6, 12.

This is an excess number, because

1 + 2 + 3 + 4 + 6 = 16 (16> 12)

No wonder that the number 12 is used in a huge number of practical areas, starting with religion: 12 gods in the Greek pantheon and the same in the pantheon of Scandinavian gods, not counting Odin, 12 disciples of Christ, 12 steps of the wheel of Buddhist samsara , 12 .d The twelve-decimal number system is one of the most convenient in practice, so it is used in the calendar to divide the year into 12 months and 4 seasons, and also to divide the day and night into 12 hours. The day consists of 2 circles of an hour hand in a circle divided into 12 segments; By the way, the amount of 60 minutes is also not without reason chosen - this is another anti-simple number with a large number of dividers.

A convenient twelve-fold system is used in several monetary systems, including in the Old Russian principalities (12 polushki = 1 altyn = 2 Ryazanki = 3 Novgorod = 4 Tver money = 6 tits). As you can see, a large number of dividers is a critical quality in conditions when coins from different systems need to be reduced to the same denomination.

Large excess numbers are useful in other areas. For example, take the number 5040. This is in some sense a unique number, here are the first from the list of its dividers:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ...

That is, the number 5040 is divided into all prime numbers from 1 to 10. In other words, if we take a group of 5040 people or objects, then we can divide it into 2, 3, 4, 5, 6, 7, 8, 9 or 10 equal groups. This is just a great number. Here is the full list of dividers 5040:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040

Damn it, yes we can divide this number into almost anything. He has 60 dividers !

5040 is the ideal number for urbanism, politics, sociology, etc. This was noticed by the Athenian thinker Plato 2300 years ago. In his fundamental work “Laws” Plato wrote that in an ideal aristocratic republic there should be 5040 citizens, because such a number of citizens can be divided into any number of equal groups of up to ten, without exception. Accordingly, in such a system it is convenient to plan the management and representative hierarchy.

Of course, this is idealism and utopia, but the use of the number 5040 is really extremely convenient. If there are 5040 inhabitants in a city, then it is convenient to divide it into equal areas, plan a certain number of service facilities for an equal number of citizens, choose representative bodies on the ballot.

Such highly complex, extremely redundant numbers are called "anti-simple". If we want to give a clear definition, then we can say that an anti-prime number is such a positive integer that has more divisors than any integer number is smaller than it.

According to this definition, the smallest anti-prime number except the unit will be 2 (two dividers), 4 (three dividers). Following are:

6 (four dividers), 12 (six dividers), 24, 36, 48, 60 (the number of minutes per hour), 120, 180, 240, 360 (the number of degrees in a circle), 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400

These numbers are conveniently used in board games with cards, chips, money, etc. For example, they allow you to distribute the same number of cards, chips, money for a different number of players. For the same reason, it is convenient to use them to compose classes for schoolchildren or students — for example, to divide them into an equal number of identical groups for the performance of tasks. For the number of players in a sports team. For the number of teams in the league. For the number of residents in the city (as already mentioned above). For administrative units in the city, region, country.

As can be seen from the examples, many of the anti-prime numbers are already de facto used in practical devices and number systems. For example, the numbers 60 and 360. This was quite predictable, given the convenience of having a large number of dividers.

One can argue about the beauty of anti-prime numbers. If prime numbers are incontestably beautiful, then anti-simple numbers may seem disgusting to someone. But this is a superficial impression. Let's look at them from the other side. After all, the foundation of these numbers are prime numbers. It is from the prime numbers, as if from building blocks, that the composite numbers are made up, the redundant numbers and the crown of creation are anti-simple numbers.

The main theorem of arithmetic asserts that any composite number can be represented as the product of several simple factors. For example,

30 = 2 × 3 × 5
550 = 2 × 5 ² × 11,

In this case, the composite number will not be divided by any other prime number, except for its prime factors. Anti-simple numbers, by definition, differ in the maximum product of the powers of the prime factors of which they are composed.

5040 = 2 ⁴ × 3 ² × 5 × 7

Moreover, their prime factors are always consecutive primes. And the degrees in the series of prime factors never increase.

So anti-prime numbers also have their own special beauty.