Prime number (or just *prime*) is a whole positive number only divisible by 1 and itself, except for the number 1. I.e. prime numbers are 2, 3, 5, 7, 11, 13, 17 etc. Prime numbers are extremely important, interesting and mysterious for their properties and distribution among other numbers, they have for millennia fascinated mathematicians, nowadays they are studied in the math subfield called number theory. Primes are for example essential in assymetric cryptography. Primes can be seen as the opposite of highly composite numbers (also antiprimes, numbers that have more divisors than any lower number).

The largest known prime number as of 2022 is 2^82589933 - 1 (it is so called Mersenne prime, i.e. a prime of form 2^N - 1).

Every natural number greater than 1 has a unique **prime factorization**, i.e. a set of prime numbers whose product it is. For example 75 is a product of three primes: 3 * 5 * 5. This is called the *fundamental theorem of arithmetic*. Naturally, each prime has a factorization consisting of a single number -- itself -- while factorizations of non-primes consist of at least two primes. To mathematicians prime numbers are what chemical elements are to chemists -- a kind of basic building blocks.

**Why is 1 not a prime?** Out of convenience -- if 1 was a prime, the fundamental theorem of arithmetic would not hold because 75's factorization could be 3 * 5 * 5 but also 1 * 3 * 5 * 5, 1 * 1 * 3 * 5 * 5 etc.

The unique factorization can also nicely be used to encode multisets as numbers. We can assign each prime number its sequential number (2 is 0, 3 is 1, 5 is 2, 7 is 3 etc.), then any number encodes a set of numbers (i.e. just their presence, without specifying their order) in its factorization. E.g. 75 = 3 * 5 * 5 encodes a multiset {1, 2, 2}. This can be exploited in cool ways in some cyphers etc.

When in 1974 the Arecibo radio message was sent to space to carry a message for aliens, the resolution of the bitmap image it carried was chosen to be 73 x 23 pixels -- two primes. This was cleverly done so that when aliens receive the 1679 sequential values, there are only two possible ways to interpret them as a 2D bitmap image: 23 x 73 (incorrect) and 73 x 23 (correct). This increased the probability of correct interpretation against the case of sending an arbitrary resolution image.

**There are infinitely many prime numbers**. The proof is pretty simple (shown below), however it's pretty interesting that it has still not been proven whether there are infinitely many twin primes (primes that differ by 2), that seems to be an extremely difficult question.

Euklid's proof shows there are infinitely many primes, it is done by contradiction and goes as follows: suppose there are finitely many primes *p1*, *p2*, ... *pn*. Now let's consider a number *s* = *p1* * *p2* * ... * *pn* + 1. This means *s* - 1 is divisible by each prime *p1*, *p2*, ... *pn*, but *s* itself is not divisible by any of them (as it is just 1 greater than *s* and multiples of some number *q* greater than 1 have to be spaced by *q*, i.e. more than 1). If *s* isn't divisible by any of the considered primes, it itself has to be a prime. However that is in contradiction with the original assumption that *p1*, *p2*, ... *pn* are all existing primes. Therefore a finite list of primes cannot exist, there have to be infinitely many of them.

**Distribution and occurrence of primes**: the occurrence of primes seems kind of """random""" (kind of like digits of decimal representation of pi), without a simple pattern, however hints of patterns appear such as the Ulam spiral -- if we plot natural numbers in a square spiral and mark the primes, we can visually distinguish dimly appearing 45 degree diagonals as well as horizontal and vertical lines. Furthermore the **density of primes decreases** the further away we go from 0. The *prime number theorem* states that a number randomly chosen between 0 and *N* (for large *N*) has approximately 1/log(N) probability of being a prime. **Prime counting function** is a function which for *N* tells the number of primes smaller or equal to *N*. While there are 25 primes under 100 (25%), there are 9592 under 100000 (~9.5%) and only 50847534 under 1000000000 (~5%).

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```

*Ulam spiral: the center of the image is the number 1, the number line continues counter clockwise, each point represents a prime.*

Here are prime numbers under 1000: 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.

```
______/
/ /
_____ ______/_ /
____ / X /__\ /
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/ \ / /\ \ / \ / \ /\
--2--3--/--5--X--7--/--\--X--11-X--13-X--\--
\__\/ \ \/ \__\/ \ \/ \__\/ \ \/ \__\/
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```

**Primality test**: testing whether a number is a prime is quite easy and not computationally difficult (unlike factoring the number). A naive algorithm is called *trial division* and it tests whether any number from 2 up to the tested number divides the tested number (if so, then the number is not a prime, otherwise it is). This can be optimized by only testing numbers up to the square root (including) of the tested number (if there is a factor greater than the square root, there is also another smaller than it which would already have been tested). A further simple optimization is to to test division by 2, 3 and then only numbers of the form 6q +- 1 (other forms are divisible by either 2 or 3, e.g 6q + 4 is always divisible by 2). Further optimizations exist and for maximum speed a look up table may be used for smaller primes. A simple C function for primality test may look e.g. like this:

```
int isPrime(int n)
{
if (n < 4)
return n > 1;
if (n % 2 == 0 || n % 3 == 0)
return 0;
int test = 6;
while (test <= n / 2) // replace n / 2 by sqrt(n) if available
{
if (n % (test + 1) == 0 || n % (test - 1) == 0)
return 0;
test *= 6;
}
return 1;
}
```

Sieve of Eratosthenes is a simple algorithm to find prime numbers up to a certain bound *N*. The idea of it is following: create a list of numbers up to *N* and then iteratively mark multiples of whole numbers as non-primes. At the end all remaining (non-marked) numbers are primes. If we need to find all primes under *N*, this algorithm is more efficient than testing each number under *N* for primality separately (we're making use of a kind of dynamic programming approach).

**Prime factorization**: We can factor a number by repeatedly brute force checking its divisibility by individual primes and there exist many algorithms applying various optimizations (wheel factorization, Dixon's factorization, ...), however for factoring large (hundreds of bits) primes there exists no known efficient algorithm, i.e. one that would run in polynomial time, and it is believed no such algorithm exists (see P vs NP). Many cryptographic algorithms, e.g. RSA, rely on factorization being inefficient. For quantum computers a polynomial ("fast") algorithm exists, it's called Shor's algorithm.

Prime generation: TODO

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