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Miloslav Ciz 2023-12-31 01:01:53 +01:00
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prime.md
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@ -19,7 +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. Another simple but unproven conjecture about prime numbers if [Goldbach's conjecture](goldbachs_conjecture.md) stating that every even number greater than 2 can be written as a sum of two primes.
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.