why is this kind of thing interesting? my undergrad math teachers were never able to convey that
is there something here to use this knowledge with? like cracking a lotto’s RNG by knowing a second prime probability? that would be interesting to me
is there something here to use this knowledge with? like cracking a lotto’s RNG by knowing a second prime probability? that would be interesting to me
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)
> "Obvoiusly" there's no uniform distribution on the natural numbers, so what does it even mean to choose a "random" one? The way the number theorists usually solve this problem is by fixing a large number N and looking at the probabilities when you pick a random number between 1 and N. Then you look at the N → ∞ limit of these probabilities.
And by using the λ₂(p) function, we can find the asymptotic proportion that each prime p appears in the data set: if the set has N items, then approximately λ₂(p)·N of them will be equal to p, and the actual number will asymptotically approach this approximation. Given this, we can look at the set in terms of the proportions of its elements. We have first, P(p < 37) = λ₂(2) + ... + λ₂(31) ≈ 0.49061; second, P(p = 37) = λ₂(37) ≈ 0.00964; and third, P(p > 37) = 1 − P(p < 37) − P(p = 37) ≈ 0.49975.
Thus, if we sort the set with N data points in ascending order, we'll ultimately see 0.00964·N instances of 37, flanked to the left by 0.49061·N values less than 37, and to the right by 0.49975·N values greater than 37. For N odd, the median is defined as the value at the 50% mark in the sorted set, which is clearly 37. For N even, the median is defined as the mean of the two values surrounding the 50% mark; since 0.00964·N → ∞ as N → ∞, there will eventually be at least two instances of 37 around the 50% mark, so the median will once again be 37. Thus, for N even and odd, the median will always end up at 37.
Many much simpler and more intuitive statements about primes and their distribution are beyond the grasp of modern math to prove or disprove.
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.
for the second kind of people (like me) these "interesting facts" are fascinating even in otherwise uselessness