# Prime numbers

Mathematics since ancient times have evolved and created new developments for its operation and ease of putting it into practice. The **prime numbers** are merely applying algorithms to the composition of composite numbers, since the findings of two divisors of the same number are based.

## What are prime numbers?

According to the mathematician **Eratosthenes of Cyrene** , prime numbers do not follow a precise logic, rather they are a product of the division in which only two divisors are found, the same number and the number (1), as he once said: *“easier it is multiplying, but finding the divisor is complex ”.*

- Definition
- features
- History
- What are they for
- How to find prime numbers
- How it differs from composite numbers
- Even prime numbers
- List of prime numbers from 1 to 2000
- Other examples
- The sum of two prime numbers is a prime number

## Definition

It is a number that is **described** as a **natural number** and they are greater than (1), since it leads to only being divided between it and the unit (1), they are based on the composite numbers, since they are governed by the division of the themselves.

## features

Among the characteristics that define these numbers as unique, they are:

- Be able to be divided between only two dividers.
- The result of the division is even.
- There are more prime numbers between 1 and 100 than between 101 and 200.
- All prime numbers, except (2), are odd numbers.
- The (1) is not a prime number, because it has only one divisor which is itself.
- Numbers that acquire more than two divisors are called composite numbers.

## History

The course of prime numbers has happened throughout the evolution of the human being, going through three epochs in history , of which the following stand out:

#### The pre-Hellenic East

The notches present in the Ishango bone, which dates back more than 20,000 years (prior to the first appearance of the writing) and which was found by the archaeologist **Jean de Heinzelin de Braucourt** , describe the isolation of four prime numbers: 11, 13, 17 and 19.

Some archaeologists interpret this fact as the first finding of the existence of prime numbers. That said, there are very few discoveries that allow us to discern the knowledge that man really had at that time.

Numerous dry clay **tablets** attributed to the civilizations that emerged in Mesopotamia throughout the **2nd millennium BC** . they show the solving of arithmetic problems and testify to the facts of the time.

The calculations required knowing the inverses of the natural ones, which have also been found on tablets, the **sexagesimal system** was used by the **Babylonians** to write these numbers.

The mathematical knowledge of the Babylonians required a solid understanding of the multiplication, division, and factorization of the natives.

#### Ancient Greece

The first proof in which we have knowledge of the prime numbers dates back to around 300 BC. C. and is in the **elements **** of Euclides** (volumes VII to IX). Euclid defines prime numbers, this shows that there are infinities of them, he expresses the greatest common divisor and the least common multiple and provides a method to find them, today it is known as **Euclid’s algorithm** . The Elements also have the fundamental **theorem** of **arithmetic** and the way to construct a perfect number from a **Mersenne** prime **number** .

**The ****Eratosthenes**** sieve** , carried out by Eratosthenes of Cyrene, is a simple method that allows finding prime numbers. Currently, the largest primes that are found, with the help of computers, use other faster and more complex algorithms.

#### From the time of the Renaissance:

After Greek mathematics , there was little advance in the study of prime numbers until the seventeenth century. In the year 1640; **Pierre de Fermat** established **Fermat’s** Little **Theorem** , later proved by **Leibniz** and **Euler** .

It is possible that a special case of this theorem was known much earlier in China .

In 1859 on the **zeta function they** described the path that would lead to the proof of the prime number theorem, it was then that **Hadamard** and **De la Vallée-Poussin** , each one separately, gave appearance to this scheme and managed to present the theorem in 1896.

During the 19th century, algorithms were created to know if a number is prime or not, factoring completely the next number (p + 1) or the previous one (p-1). Within the first case is the **Lucas-Lehmer test** , developed from 1856 on.

Within the second case is the **Pépin test** for the Fermat numbers in 1877. The general case of the primality test when the immediately preceding number is completely factored is called the **Lucas test** .** **

Starting in the 1970s, several researchers discovered algorithms to calculate whether a number is prime or not with **sub-exponential** complexity , which allows tests to be carried out on numbers of thousands of digits. Even the complex structuring of algorithms in the most powerful machines of our day for the search and ease of finding the prime numbers.

## What are they for

Currently, prime numbers have served us in the field of computing , since with them we can execute complex security codes, pattern keys, decryption of algorithms and complex calculations.

## How to find prime numbers

They can be found with the factorization of both a natural and a complex number, in which only when executing a division, two divisors can be found, that is, on the one hand, divide with itself and another, by the unit (1); the result of this division has to be an even number.

## How it differs from composite numbers

The most characteristic difference is that when dividing a prime number, it can only choose two divisors and the numbers composed by the division of numbers that are greater than (1), that is, natural numbers and the division that it carries with itself.

## Even prime numbers

In the infinity of numbers and the art of mathematics, only in prime numbers, the number two (2), is authentically, an even number and a prime number at the same time.

## List of prime numbers from 1 to 2000

#### Up to 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

#### Up to 100

53, 59, 61, 67, 71, 73, 79, 83, 89, 97

#### Up to 150

101, 103, 107, 109, 113, 127, 131, 137, 139, 149

#### Up to 200

151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199

#### Up to 300

211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

#### Up to 500

307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499

#### Up to 1 000

503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

#### Up to 2 000

1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367 , 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549 , 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723 , 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931 , 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999

## Other examples

#### Subtraction of prime numbers:

13–5 = 8 (composite number)

13–2 = 11 (prime number)

23–2 = 21 (composite number)

37–7 = 30 (composite number)

43–2 = 41 (prime number)

#### Multiplication of prime numbers:

2 x 3 = 6

11 x 3 = 33

29 x 5 = 145

17 x 7 = 119

13 x 11 = 143

#### Division of prime numbers:

11/11 = 1

11/1 = 11

89/89 = 1

89/1 = 89

41/41 = 1

41/1 = 41

## The sum of two prime numbers is a prime number

It is a yes and no, because it may be that the result of a composite number or a prime number when executing it, one of the most pertinent examples are:

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

11 + 2 = 13 (prime number)

3 + 2 = 5 (prime number)

7 + 2 = 9 (composite number)

13 + 5 = 18 (composite number)

5 + 2 = 7 (prime number)