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Miloslav Ciz 2022-05-30 17:53:44 +02:00
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prime.md
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@ -10,11 +10,13 @@ Every natural number greater than 1 has a unique **prime factorization**, i.e. a
The unique factorization can also nicely be used to encode [multisets](multiset.md) 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](cypher.md) etc.
When in 1974 the Arecibo radio message was sent to space to carry a message for [aliens](alien.md), 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](twin_prime.md) (primes that differ by 2), that seems to be an extremely difficult question.
Euklid's [proof](proof.md) 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](random.md)""" (kind of like digits of [decimal](decimal.md) representation of [pi](pi.md)), without a simple pattern, however hints of patterns appear such as the [Ulam spiral](ulam_spiral.d) -- 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 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%).
**Distribution and occurrence of primes**: the occurrence of primes seems kind of """[random](random.md)""" (kind of like digits of [decimal](decimal.md) representation of [pi](pi.md)), without a simple pattern, however hints of patterns appear such as the [Ulam spiral](ulam_spiral.d) -- 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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@ -93,7 +95,7 @@ int isPrime(int n)
[Sieve of Eratosthenes](sieve_of_eratosthenes.md) 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](dynamic_programming.md) approach).
Prime factorization: TODO
**[Prime factorization](factorization.md)**: We can factor a number by repeatedly [brute force](brute_force.md) 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](polynomial_time.md), and it is believed no such algorithm exists (see [P vs NP](p_vs_np.md)). Many cryptographic algorithms, e.g. [RSA](rsa.md), rely on factorization being inefficient. For [quantum](quantum.md) computers a polynomial ("fast") algorithm exists, it's called [Shor's algorithm](shors_algorithm.md).
Prime generation: TODO