37, the median value for the second prime factor of an integer

is it trivial that the series of medians for the second prime factor of numbers up to N is a specific number? isn't it just as plausible that this median would increase?

As we take the limit as N → ∞, the list continually grows larger. Thus, the number of instances of 37 continually increases, so that they occupy an asymptotically constant proportion (about 0.963%) of the values in the list; only ~49.061% of the values are less than 37, and only ~49.975% of the values are greater than 37. After a certain point, for even N, there will always be two instances of 37 surrounding the 50% mark in the list, so the median will still be exactly 37, and not some other value. I wrote up a longer explanation in another comment [0].

[0] https://news.ycombinator.com/item?id=38245162

I'm curious about the first number to have absolutely nothing interesting to be said about it ;)

> It just happens to be 37 for this list

What do you mean by “this list” here? It seems to be misunderstanding the result.

Well sure - the median is just one particular percentile - not a magically special one. The existence of a median also implies that there are probably limits for every other percentile - or, indeed, that for any number, there is some in-the-limit proportion of numbers for which the second prime factor is greater than that number (and for 37 that happens to be .5).

Was just about to comment this. Wonder if anyone more mathematically inclined can see a relationship.

I don’t know that problem and don’t feel like googling it, but it’s a reasonable guess that’s because

  1/e ≈ 0.36787944

It's also the most common number picked when people are asked to think of a random number between 1 and 100.

Alef 0 numbers have 2 as their second prime

The factors of 12 are 1, 2, 3, 4, 6, 12.

The prime factors of 12 are 2, 3.

Good nitpick. If you allow repeating primes, then the numbers with 2 as their second prime would be multiples of 4. So a quarter of numbers.

An important part of reading mathematics is mentally filling in the "correct" interpretation for terms. The optimal amount of detail typically depends on the target audience. Part of that is because making notation fully specified in English is typically cumbersome: If they had written "as their second non-repeating prime", someone could come along saying "shouldn't it say 'second-smallest' or 'in increasing order' as well?". In papers, the use of such terms would typically be accompanied by a formal definition using more precise (but not fully precise, see the talk below) mathematical notation.

I mention this not to be pedantic, but because inferring the "invisible" part of mathematics (when it can be done) instead of asking humans to do it explicitly tends to be a big usability improvement in theorem provers. Andrej Bauer has a nice talk about it at https://www.youtube.com/watch?v=wZSvuCJBaFU

I nominate 120 as "most interesting integer". For obvious reasons.

Mathematicians are locked in bitter struggle between the Zeroastrians who believe additive identity blesses 0 as truly the most foundational and therefore interesting integer, while the Unitarians favor multiplicative identity and thus champion 1. Uncountable souls have been lost to the conflict between these two groups.

12

Smallest abundant number

Prime numbers are not especially interesting. E.g. all groups of prime size are cyclic. But each composite number has its own "character". The "interesting" number among primes is 2. Almost every theorem about prime numbers has to consider 2 as a special case.

Much as I appreciate the reference, I think the fact this limit exists at all and is an integer is more interesting than what the integer is.

A lot of paradoxes and confused ideas begin with "choose an integer at random" - what is the average value of number chosen between 0 and infinity, for example.

> is there an integer which qualifies to be the most interesting?

I’d say that 664571016291591957042161991109590159107314773607 and 590488317782859198927092718316232684864014739572 qualify, which are the big- and little-endian versions of ASCII “the most interesting”, respectively.

17 is well known as the most random number between 1 and 20.

One other special property prime numbers can have is being irregular. And guess what the first irregular prime number is? 37. Interesting that it appears so late because irregular prime numbers do make up around 41% of all primes. See: https://encyclopediaofmath.org/wiki/Irregular_prime_number#:....

So the most interesting prime number really depends on what you consider more interesting. If you prefer the median second prime factor, then 37 is the best, but if you prefer the first irregular prime number then 37 is also the best. So it all depends on your perspective.

Another reason to like 37 is that it ends in a 7 so that it "sounds random" when someone asks for a number and it is better than 27 because it is prime. 7 is too low and 17 has connotations with bad luck. Although 37 is quite scary too: it's not just prime, which is pretty irregular, but it's also an irregular prime.

I nominate 60. Babylonian mathematics was sexagesimal (base 60) which is a superior mathematical system to base 10. The number 60, a superior highly composite number, has twelve factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many common mathematical operations are much simpler than in base 10. Base 10 has no special mathematical properties apart from as an accident of evolution, we primates ended up with 10 fingers.

We still have some vestiges of the babylonian system however, 60 minutes in an hour. 360 (6 sixties) degrees in a circle.

Historically, I'd say 60, because relative to it's size, it has many divisors (12), and among them a lot of useful ones (2, 3, 4, 5, 10), which is why our time system and trigonometry are probably based on it (360 = 6*60; 360 has 24 divisors).

I love the number 27 3^3 2+7 = 3+3+3

37 is also the smallest base (or second int argument) not supported by itoa(). Surely not a coincidence. ;)

The most interesting integer out there must certainly be the least interesting integer!

Every prime is odd and 2 is the oddest of them all!

Two is the oddest prime.

what is interesting to me is that we call imaginary integer numbers "Gaussian Integers", but we call the quaternion integer numbers "Hurwitz Quaternions" instead of "Hurwitz Integers".

I love that there could never be a least interesting natural number, because that fact would be interesting. In fact there can be no uninteresting natural numbers, because then you could list them, and the lowest one would be interesting for that fact.

In analytic number theory we usually only care about growth rates in a very coarse sense- up to scaling by some constant, and asymptotically. Because the log of any base is precisely the same up to constants, it doesn't actually matter. If you look at the expression, to the right of the equals sign is a big O- that's the same big O as the one you might be familiar with in complexity.

That being said, in math more generally when we need a concrete log the natural log is pretty much always the way to go- I haven't seen ln in a little while.

Also deep dive in the 73 and 37 https://www.youtube.com/watch?v=4DQndnAhdxk

I really dislike base-10 tricks. They are just cute, rarely have any deeper significance.

Given that RSA primes are usually checked against some probabilistic primality test like Miller-Rabin, I would think the answer is no

They grow very quickly. The median third and fourth primes are 42719 and 5737850066077 - see https://oeis.org/A284411 .

If p_k^* is the median kth prime then we have

log log p_k^* = k - C + O(1/sqrt(k))

where C is a constant around 0.59483; that is, p_k^* grows like exp(exp(k)). This is "Corollaire 1.2" of the paper that the post links to (https://tenenb.perso.math.cnrs.fr/PPP/p_k%28n%29.pdf). (It's in French.)

An unknown thing becomes a non obvious thing, which makes it interesting. Until it becomes obvious, of course.

there are people that think math is a chore - and then there are people that think math is a game

for the second kind of people (like me) these "interesting facts" are fascinating even in otherwise uselessness

I mean this is a crazy thing to know about 37. And we know it!

The interesting thing is that this feels like a thing that it would be extremely hard to know, and yet we found a way to show it to be true.

Many much simpler and more intuitive statements about primes and their distribution are beyond the grasp of modern math to prove or disprove.

I thought it'd be infinite or something. So it's cool to know it exists and it's not 1, 2 or infinity.

I don’t think this is _that_ interesting because the probabilities don’t add up to exactly 50%.

We have:

- the smallest prime factor of less than half of all integers is ≤ 31

- the smallest prime factor of over half of all integers is ≤ 37

So, clearly, the prime p where exactly half of all integers have something ≤ p as their smallest prime factor must lie between 31 and 37 :-)

Alternatively, one could argue such a prime doesn’t exist. That, I think, is the better answer.

That doesn’t imply there’s no good definition of that median, though because it’s 100% acceptable to have a set of numbers whose median is not an element of that set. https://en.wikipedia.org/wiki/Median#Finite_data_set_of_numb...:

“If the data set has an even number of observations, there is no distinct middle value and the median is usually defined to be the arithmetic mean of the two middle values.[1][2] For example, this data set of 8 numbers

1, 2, 3, 4, 5, 6, 8, 9 has a median value of 4.5”

So i guess the hunt still is on for a good argument as to why a non-prime such as 36 or even a real such as 36.716… should be called the median of the smallest prime factors of all integers (I wouldn’t know whether that exists)

Very interesting question. I thought a lot why some math facts seem boring to me while others are interesting, sometimes exciting.

I believe interesting math facts are interesting because they clash with intuition. Like you expect some freedom for a second prime factor to be anything it likes, but no, it has a very small median.

I misunderstood the claim at first, counted primes in reverse order and was really surprised by an idea that second largest prime factor is so small and started to think of Eratosthenes but continued to read. When I discovered my mistake and then it came to Eratosthenes it got almost obvious and not interesting really, I didn't even read it when it got techical. Though if not my disappointment I'd probably found it interesting, because while I believe I could find 37 as a median and prove it without any hints, I would need a nudge: I needed to be told that median of a second prime factor is a small number.

I believe that the practice of noticing these confusions and resolving it is a basis of a mathematician's mind. As a mathematician you need a highly tuned intuition that matches real theorems, because you start with intuition, formalize it as a theorem and then you look for a proof.

Intuition guesses theorems to prove or to look in literature. But if my intuition can be surprised by some fact it means it cannot guess it. And probably it cannot guess a lot of other related facts.

you assume practicality in higher mathematics? thats like ancient Babylonians complaining about the impracticality of "calculus"

That would be an incredibly expensive way of generating prime numbers, far more expensive than just checking to see if it has any factors.

There shouldn't be a direct association, are you referring to something specific? Choosing a good password is about making it less predictable.

Prime numbers do factor into cryptography though, they use prime numbers for key generation and key exchange.

there is no such thing as a good password. I have heard of good SSO's though

Probably not. We are talking about some function of first N numbers and its limit when N approaches infinity. There is more than one reasonable way to clarify "first prime factor" and "first N numbers", but I think for all of these reasonable definitions the median diverges, having different subsequential limits for even and odd values of N.

How so?

1 has no first prime factor so must be excluded along with 0. 2 is always >=50% of the head sequences of the rest:

2 3 2 5 2 7 ...

I don't think that's right. My reading of it was that 37 is the median prime among all primes.

> it's all primes up to 37

Can you elaborate on "it" in this sentence? What is "all primes up to 37", when you phrase things in an easier to understand way?

I can't tell if you're suggesting the word median is being used in a misleading way? But it's basically the definition of "median 37" that half your numbers are "up to 37".

It's very unlikely that they are related. 1/e which is approximately 0.37 is the solution to the problem you're referring to but the occurrence of 37 depends on the choice of base. It just happens to be the case that in base 10 we find that round(1/e*10^2)=37 but it would be different in other bases.

Meanwhile, the median value for the second prime is completely independent of the choice of base.

it's more that when you filter out 37-and-below, the "rest" category as a whole takes ~50% of the numbers

No, this is not an empirical result. The interesting thing is that this is a fixed number (and that it's quite small). ("After all, about half of *all* integers have 2 as their smallest prime factor" is the beginning of the intuition needed I guess)

nothing worse then when a good joke goes unnoticed.

42 must mean both 41 and 43 taken together. these two primes are a twin prime above 37. and 29 and 31 both take together mean 30. the twin pair below 37

I find it very odd that wikipedia exists up to and including 11, but then does not exist at 12. Any reason for this?