Prime numbers library
 

I was browsing through wikipedia and I was surprised, how [URL="https://en.wikipedia.org/wiki/List_of_prime_numbers#Lists_of_primes_by_type"]many types[/URL] of primes is there.

But it was hard for me to figure out, which number belongs into which type. So, I have decided to build [URL="https://prime-numbers.info/"]library of prime numbers[/URL]. It's still work in progress. I have underestimated, how many types of primes is there. :)

While I am still buildint it, I have few questions:[LIST]
There are palindromic primes, two sided primes, but there are no [URL="http://www.numbersaplenty.com/set/plaindrome/"]plaindrome/katadrome[/URL] primes. Why they aren't also interesting?
There are inconsistencies between wikipedia and OEIS. For example [URL="https://en.wikipedia.org/wiki/Prime_triplet"]prime triplets[/URL] - first member should not be there. What is authoritative source of truth?
Some of the primes have generalized forms - are they also interesting?[/LIST]
If you have any sugguestions, how to improve it, please, let me know. I will try to incorporate it.

I have a list here
[url]https://oeis.org/wiki/User:Charles_R_Greathouse_IV/Tables_of_special_primes[/url]
which may be of interest. My list is set up essentially as a sieve, removing the predictable (e.g., Diophantine) types and boring (base-dependent) ones to end up with a list of the potentially-more-interesting "other" types.

Where possible I've tried to give their density, to give an idea of how they grow. You get at this same sort of thing by listing initial members.

I also have many references if there are questions about the types, to the OEIS and the research literature.

[QUOTE=Martin;465859]There are palindromic primes, two sided primes, but there are no [URL="http://www.numbersaplenty.com/set/plaindrome/"]plaindrome/katadrome[/URL] primes. Why they aren't also interesting?[/QUOTE]

"Plaindrome" primes are [url=https://oeis.org/A028864]A028864[/url], "metadrome" primes are [url=https://oeis.org/A052015]A052015[/url], "nialpdrome" primes are [url=https://oeis.org/A028867]A028867[/url], and "katadrome" primes are [url=https://oeis.org/A052014]A052014[/url]. They were all added to the OEIS around 1999. Generally, I don't find base-dependent sequences interesting -- these numbers wouldn't look special to someone writing them in base 3 or base 16.

[QUOTE=Martin;465859]There are inconsistencies between wikipedia and OEIS. For example [URL="https://en.wikipedia.org/wiki/Prime_triplet"]prime triplets[/URL] - first member should not be there. What is authoritative source of truth?[/QUOTE]

Usually the answer is that there is no universal source of truth, different sources define things differently and you just need to be careful about which definition you're using. You can find, for example, "Powerful numbers, definition (1)" in the OEIS.

But in this case, the OEIS is right and Wikipedia is wrong. That first triple doesn't match the definition they give, which is indeed the standard definition. I'll correct the article.

[QUOTE=Martin;465859]Some of the primes have generalized forms - are they also interesting?[/QUOTE]

Sure, probably.

[QUOTE=Martin;465859]If you have any sugguestions, how to improve it, please, let me know. I will try to incorporate it.[/QUOTE]

I'd like to see more information in the entries. For mine I picked density, but you might find other things more interesting.

Thanks.

ad base dependent: I was wondering too, what is the best approach. Lets say, that I would generate them for all bases till 16. I was thinking, that it would inflate number of properties for small numbers (e.g. 3 is palindrom in any base). So I have decided for base 10 for now.

[QUOTE=Martin;465859]But it was hard for me to figure out, which number belongs into which type. So, I have decided to build [URL="https://prime-numbers.info/"]library of prime numbers[/URL]. It's still work in progress. I have underestimated, how many types of primes is there. :)[/QUOTE]

I think the hard part is there's so many sub-types of different types mersenne primes are the base 2 repunit primes. there's any number of types twin primes, cousin primes, sexy primes, cyclops primes, repunit primes, fermat primes, cullen primes, woodall primes and some can be related to each other or have overlap or be a subset of each other.

[QUOTE=science_man_88;465921]I think the hard part is there's so many sub-types of different types mersenne primes are the base 2 repunit primes. there's any number of types twin primes, cousin primes, sexy primes, cyclops primes, repunit primes, fermat primes, cullen primes, woodall primes and some can be related to each other or have overlap or be a subset of each other.[/QUOTE]

My first step is to populate some minimal dataset.
And then I will try to figure out, how to present them. One possibility is to introduce possibility to check some prime types - and it would show which numbers belong to this class.

Now I have introduced [URL="https://prime-numbers.info/special/number-cloud"]number cloud[/URL] to visualize into how many classes top numbers belong.

[QUOTE=Martin;465928]My first step is to populate some minimal dataset.
And then I will try to figure out, how to present them. One possibility is to introduce possibility to check some prime types - and it would show which numbers belong to this class.

Now I have introduced [URL="https://prime-numbers.info/special/number-cloud"]number cloud[/URL] to visualize into how many classes top numbers belong.[/QUOTE]

I think you'll see mostly [URL="https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers"]low numbers show up as large [/URL] simply because they can be in some many sequences ( when they start small) though my link is in respect to sequences.

[QUOTE=Martin;465918]ad base dependent: I was wondering too, what is the best approach. Lets say, that I would generate them for all bases till 16. I was thinking, that it would inflate number of properties for small numbers (e.g. 3 is palindrom in any base). So I have decided for base 10 for now.[/QUOTE]

I agree, I certainly wouldn't just add the properties for all those bases. If anything that would just make the situation worse, adding more base-dependent noise to your nice lists. :smile:

Maybe you could have some kind of implicit weight where some properties (Wieferich primes) are weighted heavily, while others (palprimes) are weighted less heavily. You could use this for sorting properties on the individual number pages (more weight = higher) as well as on the number cloud (more weight adds more to the size).

[QUOTE=science_man_88;465930]I think you'll see mostly [URL="https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers"]low numbers show up as large [/URL] simply because they can be in some many sequences ( when they start small) though my link is in respect to sequences.[/QUOTE]

Agreed -- though I don't think this is necessarily a bad thing. It's probably a bit comforting to see old favorites like 5 and 11 showing up large and near the center.

But if you wanted to change it -- or you wanted to offer the option to change it -- you could scale numbers not by their weight (number of properties, or sum of properties * weight if you're scaling as I mentioned above) directly but by their weight and size. For example, you might want font size to be proportional to sqrt(log(p) * w) where w is the weight and p is the number being considered. So a number twice the length with half the number of properties would be the same size.

[QUOTE=CRGreathouse;465932]Agreed -- though I don't think this is necessarily a bad thing. It's probably a bit comforting to see old favorites like 5 and 11 showing up large and near the center.

But if you wanted to change it -- or you wanted to offer the option to change it -- you could scale numbers not by their weight (number of properties, or sum of properties * weight if you're scaling as I mentioned above) directly but by their weight and size. For example, you might want font size to be proportional to sqrt(log(p) * w) where w is the weight and p is the number being considered. So a number twice the length with half the number of properties would be the same size.[/QUOTE]

it also depends, on what you call a property of course. All primes greater than 3 are either 1 mod 6 or 5 mod 6 ( yeah I know... mostly just stating for the OP) edit: [url]http://oeis.org/wiki/Ludic_numbers[/url] primes could be added.

Visualization of prime numbers density
 

I have added videos visualizing density of different prime number types.[LIST][*][URL="https://prime-numbers.info/special/visual-type-comparison#prime--lucky-prime--10000"]Primve vs Lucky primes[/URL][*][URL="https://prime-numbers.info/special/visual-type-comparison#cousin-prime--twin-prime--10000"]Cousin vs Twin primes[/URL][/LIST]
What is most surprising for you?

I like to keep my prime sublist schema simple, odds and evens - any help expanding the latter list beyond its current lonely member is appreciated. Said member likes to say that the popular song is wrong - it really should be "2 is the loneliest number."

One possible suggestion would be a Monty-Python theme, where we start with Palindromic primes and supplement with Cleesedromes, Idledromes, Chapmandromes and Jonesdromes, and then later Gilliamdromes get added in a kind of embarrassed, cartoonish poor-American-cousin fashion.