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Sexy prime (wikipedia.org)
124 points by lelf 35 days ago | hide | past | web | favorite | 40 comments



Numberphile video on sexy primes: https://youtu.be/WJ12DYBuazY

This is one of my favorite youtube channels. They make videos on various math topics that are understandable by laypeople.


Is there any practical use to these classifications? It seems to me that lager primes are distributed more or less randomly, so I don't really see the point of looking for constellations like this. Am I missing something?


Depends on what you mean by practical.

One of the underlying challenges for number theory is closing the gap between the great many statements that we can easily predict on the basis of primes acting a lot like random numbers, and the statements that we know how to prove. Both the Riemann hypothesis and the twin prime conjecture are good examples.

These constellations provide more examples of the same. We can in a straightforward way rule out constellations that can only happen a finite number of times. And for those which can happen an infinite number of times, we can predict the frequency with which they will happen.

Should any such constellation happen a statistically unlikely amount given that prediction, this would be of great interest for number theorists. Unfortunately to date they have stubbornly behaved as predicted, but that doesn't mean that the effort spent searching was wasted.


Number theory is always pointless until an application is found and it becomes a separate field of research.


I guess the question is more: what makes these primes more interesting than, say, cousin (differ by 4) or twin (differ by 2) primes, or any other differ-by-n primes.


In this case, nothing that I know of. Again, that's the kind of math that is classified as "number theory". It's just pure investigation of relationships between numbers. Occasionally you gain some insight that is useful.

A lot of cryptography used to be just number theory until computers came along and were powerful enough to make use of it. How to tell if something if someting is divisible by 3. Checksums as used on credit cards. Euler's algorighm.

No, sexy primes don't really have a point other than they are identified and there is probably some unused conjecture that they are infinite in number.


Using Euclid's algorithm (not Euler's) is certainly not the easiest way of checking for divisibility by 3 - a number is divisible by 3 if and only if it's sum of digits is divisible by 3. You can repeat the process until you have one digit.


But what if you start with a binary representation?


Do it in quaternary. Add pairs of bits.


Or you can form the alternating sum of the bits, e.g. for 0b10011001 you calculate 1-0+0-1+1-0+0-1 = 0 which is divisible by three. (That's similar to the divisibility test by 11 of a number in base-10, or more generally testing if a number in base `b` is divisible by b+1)


Probably nothing. But the general topic of prime constellations, which subsumes all of these examples, is an interesting one.



That 41 Ulam Spiral is quite beautiful...


It's a specific case of a prime gap, which might help solve some open problems in the field of number theory.


To extrapolate on your point for those interested:

The analogy I'd give is from physics. Understanding the prime number structure is like understand how an arbitrarily complex 3-dimensional shape will interact with another equally arbitrarily 3-dimensional complex shapes (let's just assume rigid-body interaction here).

But it should be intuitively obvious that starting with the question you want to answer "how do arbitrarily complex shapes interact" (the analog, in our example, to "how are arbitrary primes structured") is too big an undefined question to answer directly. Maybe somebody will be able to do it, but most likely it will be solved by breaking it into smaller, incomplete, but accurate models that though comparing and contrasting (e.g. why do circles interact differently than squares) and combination (e.g. I know circles interact, I know how squares interact, I can now define a grand circle/square unification theory that describes how circles and squares interact) .

So, you break the problem down into questions like "how do circles interact?", "how do squares interact?", "how do one-dimensional shapes interact?", "2D?". By identifying subclasses of the overall uber problem it's possible to solve a hard larger problem.

Back to the primes example, each different metric for defining a relationship between primes effectively defines a new class of primes that can be probed to figure out why they act in the way they do and how they are distributed. Each class of prime is a (probably, but not necessarily) incomplete yet accurate model for how all primes operate overall.


> Am I missing something?

Probably just a sense of humor


Pretty common on HN.


Last year both my sister and I, and my mom and dad had sexy prime years. The coolest thing is that all my family of 5 had a prime number of years, pretty incredible. I wrote about it here: https://medium.com/@0x0ece/primes-twin-primes-and-my-moms-bd...


Last year my family was all prime: 5, 7, 47, and 53. Six years from now, we will be 11, 13, 53, and 59. And then 36 year from now when the kids are approaching middle age.


> In an arithmetic progression of five terms with common difference 6, one of the terms must be divisible by 5, because 5 and 6 are relatively prime. Thus, the only sexy prime quintuplet is (5,11,17,23,29); no longer sequence of sexy primes is possible.

Pretty neat.


This page contains visualization of sexy primes up to 1 million - https://prime-numbers.info/article/sexy-primes#visualization

You can also compare their density with normal primes on this page - https://prime-numbers.info/special/visual-type-comparison#se...

Best thing is that you can enjoy those videos for 11 hours. :)


Sounds like the name of an Amazon adult streaming service


Before reading the article I thought it would be about prime numbers which encode erotic images in a similar fashion to https://en.wikipedia.org/wiki/Illegal_prime . (in which case if the image depicts a minor, it would be a sexy illegal prime...)


Website displaying random pairs: http://primes.sexy


This is clickbait done right.


Interesting we call, a topic never will know as it is not too sexy. Just number even one may say, but why they have patterns.


[flagged]


67 and 73 are the sexiest primes because they straddle 69.


Well played sir, well played...


oh boi, that hurt


69 isn't prime. it's 23x3. I think this is why I'm alone.


maybe that's why we're on HN


Now earwormed to the tune of "baby shark".

Also, Mersenne Prime, Happy Prime, Lucky Prime etc...


They didn't have the guts to call a prime triplet involving a sexy prime a threeway or ménage à trois? Boo!


Came here to say that. I was going to recommend it be called a prime threesome though.


They're primes which are separated by exactly six non-prime numbers. It remains an open question whether or not there are infinitely many sexy primes.

More generally, see the article on prime gaps: https://en.m.wikipedia.org/wiki/Prime_gap


> They're primes which are separated by exactly six non-prime numbers.

That doesn't match what the article says: "prime numbers that differ from each other by six". So they're separated by five other numbers (which are not necessarily all non-primes).


Separated by five natural numbers.


5,11 pair runs counter to this definition. .


separated by is 1 less than the difference, so separated by 5 composite numbers is what you meant? i mean it's just a wording issue.


Yes, that’s what I meant. Thanks for the charitable correction.

Great to see the HN tradition of downvoting for an off by one error continues strong as well...




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