Prime numbers

מָתֵימָטִיקָה

2022

We explain what prime numbers are, their history and what their uses and applications are. Also, differences with composite numbers.

Prime numbers cannot be broken down into smaller numbers exactly.

What are prime numbers?

In math, the prime numbers are the set of natural numbers greater than 1, which can only be divided by 1 and themselves. That is, they are numbers that cannot be broken down into smaller figures exactly, and in this they differ from the rest of the natural numbers (that is, the composite numbers). This condition is known as primality.

For example, 3 is a prime number, since it can only be divided between 1 and 3, while 4 can be divided by 2. Something similar happens with 7, a prime number, but not with 8, divisible by 2 and Four.

The list of prime numbers is infinite and seems to be subject to the laws of probability, that is, its frequency of appearance does not follow strict and regular rules.

That is why prime numbers have been the object of study since ancient times by mathematicians and thinkers, many of whom have thought to find some kind of revelation or divine message in the laws of their distribution. In fact, some of the most difficult mathematical problems to solve have to do with prime numbers, such as the Riemann hypothesis and the Goldbach conjecture.

history of prime numbers

Euclid was the first to make a formal study of prime numbers.

The study of prime numbers had its beginnings in ancient times. Evidence of their knowledge has been found in civilizations long before the appearance of the writing, about 20,000 years ago, as well as on clay tablets from ancient Mesopotamia. Both the Babylonians and the Egyptians developed a powerful knowledge mathematical in which the prime numbers were contemplated.

However, the first formal study of prime numbers appeared in Ancient Greece around 300 BC. C., and it is the Items of Euclid (in his volumes from VII to IX). Around the same time, the first useful algorithm for finding prime numbers emerged, known as the Sieve of Eratosthenes.

However, it was not until the 17th century that these studies became relevant again in the West: the French jurist and mathematician Pierre de Fermat (1601-1665), for example, established in 1640 his Theorem de Fermat, and the French monk Marin Mersenne (1588-1648) devoted himself to prime numbers of the form 2p – 1, which is why they are known today as “Mersenne numbers”.

Thanks to these studies, added to those of Leonhard Euler, Bernhard Riemann, Adrien-Marie Legendre, Carl Friedrich Gauss and other European mathematicians, the first modern methods for finding prime numbers appeared in the 19th century, precursors of those that are applied today. computers scientific.

Uses and applications of prime numbers

Prime numbers have the following applications and uses:

  • In the field of numerical and mathematical studies, prime numbers are used for the study of complex numbers, through the concept of "relative primes". They are also used in the formulation of "finite bodies" and in the geometry of star polygons of n
  • In computing, the prime numbers are used for the formulation of keys by means of algorithms calculation.

Prime Number Table

Between the number 2 and the number 1013 there are 168 prime numbers, which 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
439 457 461 463 467 479 487
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 1009 1013

Difference Between Prime Numbers and Composite Numbers

As its name indicates, composite numbers are made up of two other numbers in a symmetrical and perfect way. Therefore, composite numbers can be divided by other smaller numbers and get exact results. Prime numbers, on the other hand, are divisible only by 1 and by themselves, so they are not really "composed" of other numbers, but rather constitute a singularity in themselves.

Thus, for example, the number 16 is made up of 8 (16 divided by 2), 4 (16 divided by 4) and 2 (16 divided by 8), while the number 13 is not composed of any other number, since can only be divided by 1 and itself.

Number 1

The number 1 is an exceptional case in mathematics, since today it is considered neither a prime number nor a composite number. Until the 19th century it was thought to be a prime number, even though it does not share most of the properties of prime numbers, such as the Euler function or the divisor function. The current trend, in this sense, is to exclude 1 from the list of prime numbers.

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