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prime.md
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Prime number (or just *prime*) is a [whole](integer.md) positive [number](number.md) only divisible by 1 and itself, except for the number [1](one.md). I.e. prime numbers are 2, 3, 5, 7, 11, 13, 17 etc. Prime numbers are extremely important, [interesting](interesting.md) and mysterious for their properties and distribution among other numbers, they have for millennia fascinated [mathematicians](math.md), nowadays they are studied in the math subfield called [number theory](number_theory.md). Primes are for example essential in [assymetric cryptography](assymetric_cryptography.md). Primes can be seen as the opposite of [highly composite numbers](highly_composite_number.md) (also antiprimes, numbers that have more divisors than any lower number).
```
.##.#.#...#.#...#.#...#.....#.#.....#...#.#...#.....#.....#.#.....#...
```
*Prime number positions up to 70.*
The largest known prime number as of 2022 is 2^82589933 - 1 (it is so called [Mersenne prime](mersenne_prime.md), i.e. a prime of form 2^N - 1).
Every natural number greater than 1 has a unique **prime factorization**, i.e. a [set](set.md) 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.
@ -12,7 +19,7 @@ The unique factorization can also nicely be used to encode [multisets](multiset.
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.
**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.
@ -50,6 +57,8 @@ Euklid's [proof](proof.md) shows there are infinitely many primes, it is done by
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.
Here are twin prime numbers under 1000: 3, 4, 5, 6, 11, 12, 17, 18, 29, 30, 41, 42, 59, 60, 71, 72, 101, 102, 107, 108, 137, 138, 149, 150, 179, 180, 191, 192, 197, 198, 227, 228, 239, 240, 269, 270, 281, 282, 311, 312, 347, 348, 419, 420, 431, 432, 461, 462, 521, 522, 569, 570, 599, 600, 617, 618, 641, 642, 659, 660, 809, 810, 821, 822, 827, 828, 857, 858, 881, 882.
```
______/
/ /
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There also exists a term **pseudoprime** -- it stands for a number which is not actually a prime but appears so because it passes some quick primality tests.
**Higher order primes**, also **superprimes** or prime-indexed primes, are primes that occupy prime numberth position within prime numbers, i.e. one of first higher order primes is for example number 5 because it is the 3rd prime and 3 itself is a prime. 5 is also one of second order higher primer numbers because it is 2nd first higher order prime number and 2 is a prime number. Etc. So we may generalize this concept to a prime number order *R(x)*, which says the highest order that number *x* achieves in this sense, with *R(x) = 0* meaning *x* is not prime at all. One of very high superprimes is for example number 174440041 (lowest number with *R(x) = 12*). Prime orders for numbers up to 1000 are (leaving out the ones with order 0):
2: 1, 3: 2, 5: 3, 7: 1, 11: 4, 13: 1, 17: 2, 19: 1, 23: 1, 29: 1, 31: 5, 37: 1, 41: 2, 43: 1, 47: 1, 53: 1, 59: 3, 61: 1, 67: 2, 71: 1, 73: 1, 79: 1, 83: 2, 89: 1, 97: 1, 101: 1, 103: 1, 107: 1, 109: 2, 113: 1, 127: 6, 131: 1, 137: 1, 139: 1, 149: 1, 151: 1, 157: 2, 163: 1, 167: 1, 173: 1, 179: 3, 181: 1, 191: 2, 193: 1, 197: 1, 199: 1, 211: 2, 223: 1, 227: 1, 229: 1, 233: 1, 239: 1, 241: 2, 251: 1, 257: 1, 263: 1, 269: 1, 271: 1, 277: 4, 281: 1, 283: 2, 293: 1, 307: 1, 311: 1, 313: 1, 317: 1, 331: 3, 337: 1, 347: 1, 349: 1, 353: 2, 359: 1, 367: 2, 373: 1, 379: 1, 383: 1, 389: 1, 397: 1, 401: 2, 409: 1, 419: 1, 421: 1, 431: 3,
433: 1, 439: 1, 443: 1, 449: 1, 457: 1, 461: 2, 463: 1, 467: 1, 479: 1, 487: 1, 491: 1, 499: 1, 503: 1, 509: 2, 521: 1, 523: 1, 541: 1, 547: 2, 557: 1, 563: 2, 569: 1, 571: 1, 577: 1, 587: 2, 593: 1, 599: 3, 601: 1, 607: 1, 613: 1, 617: 2, 619: 1, 631: 1, 641: 1, 643: 1, 647: 1, 653: 1, 659: 1, 661: 1, 673: 1, 677: 1, 683: 1, 691: 1, 701: 1, 709: 7, 719: 1, 727: 1, 733: 1, 739:
2, 743: 1, 751: 1, 757: 1, 761: 1, 769: 1, 773: 2, 787: 1, 797: 2, 809: 1, 811: 1, 821: 1, 823: 1, 827: 1, 829: 1, 839: 1, 853: 1, 857: 1, 859: 2, 863: 1, 877: 2, 881: 1, 883: 1, 887: 1, 907: 1, 911: 1, 919: 3, 929: 1, 937: 1, 941: 1, 947: 1, 953: 1, 967: 2, 971: 1, 977: 1, 983: 1, 991: 2, 997: 1.
## Algorithms
**Primality test**: testing whether a number is a prime is quite easy and not computationally difficult (unlike factoring the number). A [naive](naive.md) 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](sqrt.md) (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](lut.md) may be used for smaller primes. A simple [C](c.md) function for primality test may look e.g. like this: