Update

prime.md

@@ -85,9 +85,16 @@ There also exists a term **pseudoprime** -- it stands for a number which is not
 
 **[Fun](fun.md) with primes**: thanks to their interesting, mysterious and [random](randomness.md) nature, primes can be played around -- 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 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. The above mentioned pseudoprimes, superprimes and twin primes are examples of generalizing primes, another one is e.g. defining so called **almost primes** -- a number is *n*-almost-prime if it has *n* prime factors, so 1-almost-primes are just regular primes (they have 1 prime divisor -- themselves) but then there are 2 almost primes like 9 or 15 that are kind of closer to being primes than let's say 5-almost-primes such as 48 or 80. I.e. we took the idea of numbers having either none (primes) or some (non-primes) divisors and generalized it by says a number is more prime like if it has fewer divisors. Another generalization was hinted on above: construct a [tree](tree.md) of numbers, 1 being the root, assigning each number a parent that's its greatest divisor (exclude the number itself from divisors); in this tree 1 is above prime numbers, prime numbers are on level 1, second level may be seen as the "next best thing" to primes (4, 6, 9, 10, 15, ...), third level the next (8, 12, 18, 27) and so on, i.e. we define the "primeness" as a number of times we have to replace it with its greatest divisor before we get to 1. Similarly we may try to play on this observation: a non-prime is a number that is divisible by something, i.e. there is some number that when dividing the original number gives remainder after division zero; primes are those for which no number gives remainder zero, but some primes might be considered "weaker" by giving very low or very high remainder such as 1, i.e. being "not quite but almost" divisible by something (of course we have to somehow account for the fact that low divisors can only ever give low remainders) -- ideal prime would have remainders after division near the half of the dividing number (it would dodge multiples of other numbers with some margin), which we can formalize and define kind of "prime strength".
 
-TODO: generalization to non integers? does it exist?
+**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.
+- **almost primes**: a number is *n*-almost prime if it has *n* prime factors, so 1-almost primes are just regular primes (they have 1 prime divisor -- themselves) but then there are 2 almost primes like 9 or 15 that are kind of closer to being primes than let's say 5-almost-primes such as 48 or 80. We take the idea of numbers having either none (primes) or some (non-primes) divisors and generalized it by says a number is more prime like if it has fewer divisors.
+- Another idea hinted on above: make a [tree](tree.md) of numbers with 1 as its root, assign each number a parent that's its greatest divisor (excluding the number itself); in this tree 1 is above prime numbers, prime numbers are on level 1, second level may be seen as the "next best thing" to primes (4, 6, 9, 10, 15, ...), third level the next (8, 12, 18, 27) and so on, i.e. we define the "primeness" as the depth in this tree, the number of times we have to replace the number with its greatest divisor before we get to 1.
+- **[complex](complex_number.md) (Gaussian) primes**: This is not a strict generalization because we remove some primes by were primes before, but we may define prime numbers also within complex integers. Here we get primes to be 3, 7, 11, 19, 23 etc.
+- Similarly we may try to play on this observation: a non-prime is a number that is divisible by something, i.e. there is some number that when dividing the original number gives remainder after division zero; primes are those for which no number gives remainder zero, but some primes might be considered "weaker" by giving very low or very high remainder such as 1, i.e. being "not quite but almost" divisible by something (of course we have to somehow account for the fact that low divisors can only ever give low remainders) -- ideal prime would have remainders after division near the half of the dividing number (it would dodge multiples of other numbers with some margin), which we can formalize and define kind of "prime strength".
+- TODO: generalization to non integers? haven't found anything
+- ...
 
 ## Algorithms