Prime Numbers (Meaning and Explanation)

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

What are prime numbers?

In mathematics, 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 decomposed into smaller figures exactly, and in this they differ from the rest of the natural numbers (that is, composite numbers). This condition is known as primality.

For example, 3 is a prime number, since it can only be divided by 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 4.

The list of prime numbers is infinite and appears 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 subject of study since ancient times by mathematicians and thinkers, many of whom have thought to find in the laws of their distribution some type of revelation or divine message. 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

prime numbers history euclid — 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 its knowledge has been found in civilizations long before the appearance of writing, around 20,000 years ago as well as on clay tablets from ancient Mesopotamia. Both the Babylonians and the Egyptians developed powerful mathematical knowledge in which prime numbers were contemplated.

Nevertheless, The first formal study of prime numbers appeared in Ancient Greece around 300 BC. c and it is about the Items of Euclid (in its volumes VII to IX). At that 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 his Fermat Theorem in 1640, and the monk French Marin Mersenne (1588-1648) was dedicated to prime numbers of form 2^p – 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, In the 19th century, the first modern methods for finding prime numbers appeared precursors of those used today in scientific computers.

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 cousins”. They are also used in the formulation of “finite bodies” and in the geometry of star polygons. n
In computing, prime numbers are used for key formulation using calculation algorithms.

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 their name indicates, composite numbers are made up of two other numbers in a symmetrical and perfect way. That's why, Composite numbers can be divided by other smaller numbers and obtain exact results. Prime numbers, on the other hand, are only divisible by 1 and by themselves, so that they are not really “composed” of other numbers, but rather constitute a singularity in themselves.

The number 1

The number 1 is an exceptional case in mathematics, since today It is neither considered 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 that sense, is to exclude 1 from the list of prime numbers.

Continue with: Ordinal numbers