7:45 PM
@Semiclassical Hey I have a numerical experiment question
that you may or may not be interested in
Let $S_n$ be the set of pairs of nonnegative integers adding to $2n$ (so $\{(0,2n), (1,2n-1), (2,2n-2), \dotsb, (n,n)\}$)
(not caring about the orders of the elements in the pairs)
Let $S_{n,k}$ be $\{(a,b)\in S_n\mid p_i{\not\mid}a,b,~1\le i\le k\}$
so, like, the set of pairs where neither of the entries are a multiple of any of the first $k$ primes
This is like the sieve of Eratosthenes, where at each step we're sifting out the things that are multiples of the $k$th prime
(ex: $S_{49,3}$ is the set of pairs of things adding to $98$ where neither is a multiple of $2$, $3$, or $5$)
(which is $\{(1,97),(7,91),(19,79),(31,67),(37,61),(49,49)\}$)
Also as $k\to\infty$, the set $S_{n,k}$ becomes the set of primes adding to $2n$
(plus possibly $(1,2n-1)$, if $2n-1$ is prime)
so Goldbach's conjecture says that that isn't empty, right? (And that it's still not empty when we delete $(1,2n-1)$, which is an annoying edge case)
Consider $|S_{n,k}|/|S_{n,k-1}|$
Intuitively that should be around $(p_k-2)/p_k$ or $(p_k-1)/p_k$
(About $1/p_k$ of the elements of $S_{n,k-1}$ will have the first entry be a multiple of $p_k$, and about $1/p_k$ of them will have the second entry be a multiple of $p_k$)
(and those might overlap if $2n$ itself is a multiple of $p_k$)
So the question is, how accurate is that guess
Like, if you compute $|S_{n,6}|/|S_{n,7}|$ for lots of $n$, how closely is it approximated by $15/17$ and $16/17$
The reason I'm interested in this is that, if the lower bound is reasonably close to $(p_k-2)/p_k$, this implies the Goldbach conjecture.
The reason is that $n(\frac{3-2}3)(\frac{5-2}5)(\frac{7-2}7)\dotsb(\frac{p-2}p)$ where $p$ is the largest prime less than $\sqrt{2n}$, goes to infinity
and basically is never near zero except for small $n$
so if that's a good approximate lower bound for $|S_{n,\infty}|$, then we get that there's lots of pairs of primes adding to $2n$
Apparently if we graph that approximate lower bound we get this
(Orange is $n(\frac{3-2}3)(\frac{5-2}5)(\frac{7-2}7)\dotsb(\frac{p-2}p)$ where $p$ is the largest prime less than $\sqrt{2n}$, blue is the number of pairs of primes adding to $2n$)
You'll notice that this isn't actually a perfect lower bound - for example, there are 13 pairs of primes adding to 992 and the formula predicts a lower bound of 15
(well, 14 if we ignore the pair (1,991) — 991 is prime)
so this approximation is quite rough, and this is not an actual proof of the Goldbach conjecture
but it's a good heuristic support I think
(Credit for the idea goes to a certain Reddit user who is convinced that this is a valid proof)
(but I don't really think it'll go anywhere)
I am trying to show that a non-abelian group $G$ of order $p^3$ is isomorphic to one of two groups constructed on page 48 of these group theory notes (see examples 3.14 and 3.15 on that page).
Here is some of my work so far:
Since $G$ is a $p$-group, it must have non-trivial center; but, a...
