In math, a prime number is a positive integer that has exactly two different divisors. That is, **a number is prime when it is a positive integer other than 0 and 1, and that it can only be divided by itself and by 1 to give an exact solution** . Now if a number greater than 1 has more than two divisors, then it is called a composite number.

**Prime numbers** are relevant in many areas of pure mathematics, such as algebra or geometry. Recently, prime numbers have assumed a crucial importance also in applied mathematics, and in particular in cryptography.

Index

## Characteristics of prime numbers

Among the characteristics of prime numbers, we can name:

- The numbers 0 and 1 are neither prime nor composite numbers.

- There are more prime numbers between the number 1 and 100 than there are between the number 101 and 200.

- Prime numbers are infinite. This demonstration was made by Euclid.

- The prime numbers are all odd, except the number 2 and can be divided into two groups: one group is composed of multiples of 4 except the number 1 (3, 11, 19, etc.) and another is formed by multiples of 4 including number 1 (5, 13, 17, etc.).

- A prime number is a natural number that only has two factors, which are the number itself and the number 1.

## Prime Number Applications

Prime numbers apply:

- When studying complex numbers, it is important to go to the definition of relative primes, this is to define the primitive roots of unity. If
**n**is a**prime number**, each of the roots of 1 itself are primitive roots, except for the root of 1.

- When using a finite field, the number of elements in a ring is required to be a
**prime integer**. So, in the case of the elimination of zero, it results that each element has the multiplicative inverse and thus the structure of a body is obtained.

- In the application of a star polygon with
**n**sides, to obtain the points of**m**in**m**, as a requirement the**m**has to be less than**n / 2**and**prime**with**n.**

- Conceptualizing the canonical representate a rational number, to using equivalence classes of ordered pairs of integers, obligatorily numbers, it must involve two
**numbers****prime integers.** - The RSA algorithm has a structure based on the acquisition of a public key, through the multiplication of two numbers greater than 10100 that must be
**prime**. The security of this algorithm is based on the fact that there are no accelerated ways to decompose a large number into its**prime factors**, with a conventional computer.

## Property of prime numbers

The fundamental theorem of arithmetic states that all natural numbers have a unique symbolization as a result of the prime factors, except the order. In this way, the same prime factor can be found several times. Thus the representation of the number 1 is nothing more than an empty product.

This property is important, because it is one of the logical reasons for separating 1 from the set of prime numbers. If 1 were accepted as a prime number, the statement in the theorem would require further explanation.

In this way, the property of prime factors evolves with different definitions used in the area of mathematics, and these are:

**The least common multiple:**To calculate the least common multiple of two or more numbers, each number is decomposed into its prime factors and the result is obtained by multiplying the common and uncommon multiples raised to their greatest exponent. An example of this is, the least common multiple of 12 and 10 is 60. Where 12 = 2^{2}x 3 and 10 = 2 x 5. The result is 60 = 2^{2}x 3 x 5.

**The greatest common divisor**: To calculate the greatest common divisor of two or more numbers, each number is decomposed into its prime factors and the result is obtained by multiplying the common multiples raised to their lowest exponent. An example, the greatest common divisor of 12 and 10 is 2. Where 12 = 2^{2}x 3 and 10 = 2 x 5. The result is 2.- Two or more numbers are prime to each other, if they do not have any prime factor in common, that is, their greatest common divisor is equal to 1. Any prime number is coprime with any natural number, if it is not a multiple of itself.

## Formula for calculating prime numbers

A formula for calculating prime numbers is an expression that generates only prime numbers. Exact formulas are not known to calculate them. That is, there are no formulas that use limits, nor series , nor sums, and their duration depends on the initial data, to find all the prime numbers up to n, or even the nth first.

Instead, some formulas have been found that generate only prime numbers, although they are fundamentally useless from a practical point of view.

## How to identify prime numbers with the Sieve of Eratosthenes?

To calculate the prime numbers through the Sieve of Eratosthenes, the following must be done:

- The numbers from 1 to 100 are placed in a table, where it is easy to observe the patterns that make up the multiples of each number. The 1 must be marked or rolled, as it is not considered a prime number.

- The multiples of 2 are searched for and marked, with the exception of 2, since it is known that it only has 1 and 2 as divisors, so it is prime. All of these numbers will be composite.

- Now, with the remaining ones, the multiples of 3 are found and marked, except for 3, which is prime. A simple way is to count by 3 at a time.

- Now it is the turn to look for the multiples of 5. Those of 4 would not be necessary, because all the multiples of 4 are also multiples of 2. It is easy to find the multiples of 5, since they are all those that end in 0 or 5.
- The multiples of the number 7 are searched for. The multiples of 6 do not need to be searched, since 6 = 2 x 3.

## List of prime numbers from 1 to 1000

The 168 prime numbers less than 1000 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, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 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, 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.

## Decomposition of prime numbers.

To be able to decompose composite numbers into their prime factors, the given number has to be divided by the smallest of its prime divisors, then the quotient is also divided by the smallest of its prime divisors, and so on it is done with the other quotients up to find a prime quotient that will divide by itself and this will result in quotient 1.

An example of the decomposition of prime numbers is:

**To decompose 84 into its prime factors, do the following:**

- First divide the number 84 by 2.

- Below 84 is written the quotient 42, which is the result of the previous division.

- The 42 is divided by 2 and its quotient 21 is written below it.

- Now we divide 21 by 3 and its quotient 7 will also be noted below it. The number 21 cannot be divided by 2 so it will be divided by 3.

- The number 7 is prime, so it cannot be divided by 2, by 3 or by 5.

- Therefore, it will not be divided by itself, that is, 7 by 7 and its quotient 1 will be noted below it.

- Finally, as a result, the number 84 already has a quotient of 1 and thus the factoring is finished.

- So,
**the prime factorization of the number 84 = 2 x 2 x 3 x 7 = 2² x 3 x 7**

## conclusion

To conclude, prime numbers are the fundamental basis, not only in mathematics but in everyday life, since these numbers stand out for being an aid in different areas such as computing or engineering. Nor should we forget that prime numbers are in accordance with any other existing numbering system.

Dr. Samantha Robson ( CRN: 0510146-5) is a nutritionist and website content reviewer related to her area of expertise. With a postgraduate degree in Nutrition from The University of Arizona, she is a specialist in Sports Nutrition from Oxford University and is also a member of the International Society of Sports Nutrition.