23 is a prime number

Prime numbers


It is well known that one is Prime number a natural number different from 1 that has no factors other than 1 and itself. Even in antiquity, Greek mathematicians knew that every natural number can be uniquely decomposed (except for the order of the factors) into a product of prime numbers and that there are an infinite number of different prime numbers. In Book IX of the elements of Euclid (around 300 BC) there is the following Proof of contradiction: If one assumes that there are only finitely many prime numbers, for example, then the number is greater than all these prime numbers and is not shared by any prime number. So m is itself a prime number, a contradiction to assumption.

With the help of Sieve method of Eratosthenes (around 200 BC) one can find every prime number one after the other as follows. You start with the infinitely long list of all natural numbers greater than 1. In it, the smallest number, 2, is a prime number. You remove all their multiples from the list. The smallest number in the remainder of the list that is greater than the prime number just found, in this case 3, is the next prime number. You now remove all their multiples from the list, etc.

This results in the following prime numbers from 1 to 1000:

2 35 71113171923293137
414347535961677173798389
97101103107109113127131137139149151
157163167173179181191193197199211223
227229233239241251257263269271277281
283293307311313317331337347349353359
367373379383389397401409419421431433
439443449457461463467479487491499503
509521523541547557563569571577587593
599601607613617619631641643647653659
661673677683691701709719727733739743
751757761769773787797809811821823827
829839853857859863877881883887907911
919929937941947953967971977983991997

The hunt for large prime numbers

Of course, this theoretical method is not practical to quickly generate an arbitrarily long list of prime numbers. To this day, no formula is known that allows the rapid calculation of prime numbers of any size, or the production of an even larger prime number from a given prime number.

Therefore, with the current state of research, there is always a currently largest known prime and the current record stands at

It is by no means the case that all prime numbers below this very large number are known, as would be the case with Eratosthenes' sieving process. Such a number can be determined with the help of the Mersenne numbers. So this number is itself also a Mersenne number.

Furthermore, one can also determine special relationships for certain prime numbers. So there is for example
  • Prime twins, which are two prime numbers that have a difference of two.
    Specific examples are 5 and 7, 11 and 13, 17 and 19, 29 and 31, 41 and 43, ..., where the largest known twin prime pair is 33218925 * 2 ^ 169690 +/- 1.
    To date, however, it is unknown whether there are an infinite number of prime twins.
There is also
  • Prime triplets, which always contain three prime numbers, each of which has a difference of two. Since every third odd number is divisible by three, there is only one triplet (3, 5, and 7).
There are also
  • Prime quadruplets, each characterized by two prime twins that are only four numbers apart. Examples would be: 11; 13; 17; 19 or 101; 103; 107; 109
A very practical one application find the prime numbers in cryptography, because many encryption systems, for example the RSA method, are based on the fact that large prime numbers can be found and multiplied very quickly. For example, you can easily find two 500-digit prime numbers within seconds and multiply them together. On the other hand, there is no efficient way to factor these numbers again. Even with today's methods, recovering the two prime factors from this 1000-digit product would take millions of years.

In the Prime factorization the fundamental theorem of arithmetic applies. This means that every positive whole number can be clearly represented as a product of prime numbers except for the sequence. The prime numbers appearing in this representation are called the prime factors of the number. The difficulties involved in prime factorization are known as factoring problems. But one tries to minimize them with suitable factoring methods. Since every natural number can be written as a product of prime numbers, the prime numbers occupy a special position in number theory. They therefore occupy a position similar to that of atoms in chemistry.

The Sophie Germain primes are a whole class of primes that have played an important role in certain number theoretic investigations. Another interesting class of prime numbers, which are closely related to the ability to construct regular corners using compasses and rulers alone, are the Fermat prime numbers.