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@ -18,7 +18,7 @@ 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 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 former problem, shows to be an incomparably more difficult question to answer. The concept of twin primes can be extended to prime triplets (3 primes having a gap of 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. 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.
**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.
Euklid's [proof](proof.md) shows there are infinitely many primes, it is conducted 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.