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Miloslav Ciz 2025-05-20 21:52:04 +02:00
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prime.md
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@ -16,6 +16,35 @@ 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.
```
_ _ _ _ _ _ _ _ _ _ _
1 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4
____________________________________________________________________
1 |<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
2 | <>||<>||<>||<> <>||<>||<> <>||<>||<> <>||<> <> <>||<>||<> <>
3 | <> ||<>|| <> ||<>|| <> ||<>|| <> ||<> <> ||<>|| <>
4 | <>|| ||<> ||<>|| <>|| ||<> ||<> <>|| ||<>
5 | <> || <>|| || <> || ||<> || <> ||<>||
6 | <>|| ||<>|| ||<>|| || || ||
7 | <> || ||<> || || <> || || ||
8 | <> || || <>|| || ||<> || ||
9 | <> || || ||<>|| || <> || ||
10 | <>|| || || ||<> || ||<>||
11 | <> || || || <>|| || || <>
12 | <>|| || || ||<> || ||
13 | <> || || || <> || ||
14 | <> || || || <>|| ||
15 | <> || || || ||<>||
16 | <>|| || || || ||<>
17 | <> || || || || <>
18 | <>|| || || ||
19 | <> || || ||
20 | <> || || ||
```
*Divisibility of numbers on top by numbers on the left -- we can see prime numbers as the ones avoided by all divisors (except 1 and self), i.e. the ones for which we can draw a continuous straight vertical line between the top line (divisibility by 1) and the diagonal (divisibility by self).*
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 quite elementary (shown below), however it's pretty fascinating that it has still not been proven whether there are infinitely many **[twin primes](twin_prime.md) (primes that differ by 2)**, which, despite its similarity to the original problem, shows to be an incomparably more difficult question to answer. The notion of "twin prime" can be extended to prime triplets (3 primes with gaps 2 and 4, in either order), prime cousins (primes spaced by 4), sexy primes (primes spaced by 6) and eventually generalized to so called prime *k-tuples*: tuples describing prime patterns with offsets, e.g. [0, 2, 6] specifies one possible form of a prime triplet etc. -- you may even hear the term *prime constellation* (it's almost as if prime numbers are like stars in the sky, together forming certain shapes). Another simple but unproven conjecture related to prime numbers is [Goldbach's conjecture](goldbachs_conjecture.md) stating that every even number greater than 2 can be written as a sum of two primes.
@ -81,11 +110,12 @@ There also exists a term **pseudoprime** -- it stands for a number which is not
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.
Out of autistic curiosity we may turn this into a "race" for firsts, i.e. look for first prime of order *N* for each *N*. Here we go: 2 (order 1), 3 (2), 5 (3), 11 (4), 31 (5), 127 (6), 709 (7), 5381 (8), 52711 (9), 648391 (10), 9737333 (11), ...
**Prime gaps**: statistically gaps between consecutive primes increase. The size of the gaps themselves make another number sequence that starts like this 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4, 8, 10.
**[Fun](fun.md) with primes**: thanks to their interesting, mysterious and [random](randomness.md) nature, primes can be played around with -- of course, you can examine them mathematically, which is always fun, but you can also play sort of [games](game.md) with them. For example the prime race: you make two teams of primes, one that gives 1 modulo 4, the other one that gives 3; then you go prime by prime and add points to each team depending on which one the prime falls in; the interesting thing is that team 3 is almost always in lead just by a tiny amount (this is known as Chebyshev bias, only after 2946 primes team 1 gets in the lead for a while, then at 50378 etc.). Similar thing can be done by evaluating the Mobius function: set total sum to 0, then go number by number and if it only has unique prime factors, add 1 if the number of those factors is even, otherwise subtract 1 -- see how the function behaves. Of course you can go crazy, make primes paint pictures or compose [music](music.md) -- people also like to do this with digits of numbers, e.g. those of [pi](pi.md) or [e](e.md).
**Can we generalize/modify the concept of prime numbers?** Yeah, sure, why not? The ways are many, we'll rather run into the issue of analysis paralysis -- choosing the interesting generalization of out of the many possible ways. Some possible generalizations include:
- **pseudoprimes**: the above mentioned, i.e. non-primes passing many prime tests.