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Miloslav Ciz 2024-08-27 22:53:54 +02:00
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@ -22,7 +22,7 @@ When in 1974 the Arecibo radio message was sent to space to carry a message for
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 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%).
**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.md) -- 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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