Interestingly, the numbers in mathematics are unknown

Numbers play a very important role in all areas of human life. Inventing and naming numbers follows the interesting rules.

Discover interesting things about numbers in a whole new perspective for interesting discoveries about mathematics and numbers. However, when reading, please do not try to understand, if you are not really curious, because they . quite damage the brain.

1. Pair of intimate numbers

Two numbers form a pair of intimate numbers when they follow the rule: This number equals the sum of all the divisors of the other (except for that number itself) and vice versa. The first pair of friendly numbers was found, and also proved to be the smallest "intimate" pair, the pairs of numbers: 220 and 284 . Try a little analysis: The number 220 outside itself, it also has 11 divisors of 1, 2, 4, 5, 10, 11, 20, 44, 55 and 110. The sum of these 11 divisors is just right equal 284. On the contrary, number 284 is outside of itself, it has 5 other divisors: 1, 2, 4, 71, 142, their sum is just equal to 220.

In the 17th century, French mathematician Fecma found the second "intimate number" pair: 17296 and 18416 . At the same time, another French mathematician found the third pair: 9363544 and 9437056 . The most surprising thing is that the famous Swiss mathematician O-le in 1750 announced 60 pairs of close numbers at once. Mathematicians were horrified, they said: "O-le has found it all." But unexpectedly, a century later, a 16-year-old Italian young man named Baconi announced a pair of intimate numbers in 1866, which was only slightly larger than 220 and 284, which were pairs of 1184 and 1210 . Large mathematicians had previously found them, allowing these large, large pairs to easily cross over.

Along with the development of science and technology, computer mathematicians checked all numbers within 1,000,000, a total of 42 pairs of intimate numbers were found. Currently, the number of intimate number pairs found has exceeded 1000. But is the intimate number more infinite? Are they distributed with rules? These issues are still left open.

With the current age of technology, just by a not too complicated C ++ algorithm, you can find a lot of intimate pairs.

2. Pair of betrothal numbers

Not only did they stop at intimacy, taking one step further, scientists began to define the "betrothal number".

The pair of betrothal numbers is two positive integers such that: the sum of the divisors of this number (not counting that number) is one more than the other. In other words, (m, n) is an engaged pair of numbers if s (m) = n + 1 and s (n) = m + 1, where s (n) is the floating sum of n: one thing Equivalent is that σ (m) = σ (n) = m + n + 1, where σ denotes the sum of the divisors.

The first betrothal pairs were found: (48, 75), (140, 195), (1050, 1925), (1575, 1648), (2024, 2295), (5775, 6128).

It has been proved that the betrothed pair always consists of an even number and an odd number (perhaps representing 1 male and 1 female).

3. Emirp

If you are trying to look up the word in English, you probably won't find it. Because it is the reverse writing of the word 'Prime' .

An emirp is a prime number that, when reversing the position of its digits, also gets a prime number. This definition does not include reverse primes (like 151 or 787), nor a 1-digit prime number like 7.

The first emirps found were: 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157 .

As of November 2009, the largest emirp known is 1,010,006 941,992,101 × 104,999 1, found by Jens Kruse Andersen in October 2007.

4. Perfect number

In number theory, a positive integer is called a perfect number when it is equal to all its positive integers, except itself. Or another definition, some is called perfect when it is half its total positive integers (including itself). For example, the first perfect number is 6, because: 6 = 1 + 2 + 3, or 6 = (1 + 2 + 3 + 6) /2.m

Historically, the first four perfect numbers: 6, 28, 496 and 8128 have long been known in Greek mathematics by the mathematician Nicomachus in the form of: 2n − 1 (2n - 1):

1. When n = 2: 21 (22 - 1) = 6
2. When n = 3: 22 (23 - 1) = 28
3. When n = 5: 24 (25 - 1) = 496
4. When n = 7: 26 (27 - 1) = 8128.

Note that: 2n - 1 are prime numbers in each of these examples, Euclid proves that the formula: 2n − 1 (2n - 1) will give an even number even when and only if 2n - 1 is the number element (Mersenne primes).

In a written manuscript between 1456 and 1461, an anonymous mathematician gave the fifth perfect number: 33,550,336. In 1588, Italian mathematician Pietro Cataldi identified (8589869056) and (137,438,691,328) were the perfect sixth and seventh numbers.

Euclid proved that 2n − 1 (2n - 1): is a perfect number when 2p-1 is a prime number. For 2n-1 to be a prime number, n must also be a prime number. For example: n = 2 => 2 * (2 ^ 2-1) = 6; n = 3 => 2 ^ 2 (2 ^ 3-1) = 28. The prime number of the form 2n-1 is called the Mersenne prime number , taking the name of seventeen Marin Mersenne monks, who study reason Perfect numbers and numbers. Until the 18th century, Leonhard Euler proved: 'every Mersenne element produces a perfect number, and vice versa, each number is perfect for a Mersenne prime number' . This result is often called the Euclid-Euler Theorem.

As of February 2013, 48 Mersenne primes and so 48 perfect numbers have been known. The largest of these is 257.885.160 x (257.885.161-1) with 34,850,340 digits.

5. Strong numbers

The origin of this name comes from the accumulation of Achilles heel. As a powerful war hero, there is only one weakness: the heel. Perhaps from here, the distinction is made of three terms: perfect number, Achilles number, and strong number.

A number is called a strong number when it is simultaneously divisible by a prime number and divisible by the square of that prime number. For example, number 25 is a strong number, because it has just been divisible by the prime number 5, and the square of 5 (ie 25). Thus, some strong, may also coincide with a perfect number (the perfect number is intended to be thought of as above).

Some Achilles are strong numbers, but not perfect numbers.

The following is a list of all strong numbers between 1 and 1000: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121 , 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625 , 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.

6. Odd numbers

To understand what odd numbers are, we need to go through two definitions: Rich numbers and perfect sales numbers .

Rich numbers are numbers for which the sum of the divisors of that number (excluding itself) is greater than that number. For example, the number 12 has the sum of the divisors (excluding 12) is 1 + 2 + 3 + 4 + 6 = 16> 12. Therefore 12 is a rich number.

Perfect sale numbers are natural numbers equal to the sum of all or some of its divisors. Thus, the perfect set of numbers is larger than the perfect set of numbers. Some perfect sales numbers: 6, 12, 18, 20, 24, 28, 30, 36, 40 .

Thus, between the two sets of perfect sales numbers and rich numbers there are common elements.

And finally, what's the odd number? Some are odd numbers if it's a rich number but not a perfect number. In other words, its sum of divisors is greater than that number, but the sum of some or all of the divisors is never equal to that number.

The first few numbers in odd numbers are: 70, 836, 4030, and 5830.

7. Number of happiness

Some happiness is determined by the following process:

Start with any positive integer, replace the number with the sum of the squares of its digits, and repeat the process until the number is 1 (where it will stay), or it iterates infinitely in a cycle that does not include 1.

The numbers that this process ends in 1 are happy numbers, while those that do not end in 1 are unsatisfied numbers (or sad numbers).

Let's try with number 44:

+ First, 4 ^ 2 + 4 ^ 2 = 16 + 16 = 32.

+ Next: 3 ^ 2 + 2 ^ 2 = 9 + 4 = 13.

+ And again: 1 ^ 2 + 3 ^ 2 = 1 + 9 = 10.

+ Last: 1 ^ 2 + 0 ^ 2 = 1 + 0 = 1.

That is some happiness.

The interesting thing is that happiness numbers are very common, there are 143 numbers from 0 to 1000. And the greatest number of happiness with no repeated digits is: 986,543,210. It is a true happiness figure.

8. Inviolable numbers

This odd name is given to the numbers 'impossible' written as the sum of all the divisors of any positive integer (not counting that positive integer).

For example, 4 is not an inviolable number because 4 = 3 + 1. In which 3 and 1 are all the submissions of 9. 5 is the inviolable number because the only way to write 5 = 4 + 1. If you reason that this is the sum of 4, then you are wrong. Since the sum of the divisors of 4 must be: 1 + 2 = 3.

The first inviolable numbers: 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290 .

9. Self-satisfied number

The complacent number is the number equal to the sum of the third order caps of each of its digits. Example:

153 = 1 ^ 3 + 5 ^ 3 + 3 ^ 3.
370 = 3 ^ 3 + 7 ^ 3 + 0 ^ 3.
371 = 3 ^ 3 + 7 ^ 3 + 1 ^ 3.
407 = 4 ^ 3 + 0 ^ 3 + 7 ^ 3 .

Numbers, when named by scientists, themselves recognize their frivolity. The mathematician he, GH Hardy even published in his book "Apology of Mathematics": "These are strange concepts, very suitable for puzzle columns and are capable of entertainment, but nothing attractive to mathematicians ". Anyway, please give the reader a new perspective on mathematics.