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So, to be more explicit about the geometric series approach to that summation: S(z):=\sum_{n\geq 1}n^3z^n=\left(z \frac{d}{dz}\right)^3 f(z)=zf'+3z^2f''+z^3f''' where f(z)=(1-z)^{-1} is the geometric series in the unit disk (which is easy to differentiate). This gives us the rational...

@realisenothing Yep, solvability of a polynomial equation is related to a certain group theoretic property. The symmetric group S_n has this property iff n < 5. (And A_60 fails to have this property as it is structurally pretty much the same as S_5.)

Idk how Bernoulli did it, but you can easily evaluate such sums (replace 3 with any positive integer in fact) by repeatedly applying the operator (z d/dz) to the geometric series, which we already know how to sum. Then just chuck in 1/2.

Well 1-4p has to be the negative of a Heegner number, not a Heegner number itself, but yes this is pretty cool.

Yes, I agree that j-invariants (and stuff involving modular forms in general) are a much better example than Riemann's of the interconnectedness of mathematical branches.

That's exactly the point, it led to a surprising link between quite different areas where there was not one previously. And a very useful link at that. How does that make it not an ideal example to show that nearly anything in mathematics can be interconnected? Obviously, there is only so much...

They are powerful conjectures that have many consequences, but the vast majority of these consequences are number theoretic in nature. I don't know if they are the best examples to give of the interconnectedness of mathematics. (Although single theorems rarely give a full view of this...

Okay, so some things in the last few centuries that are more significant and central to mathematics as a whole (centrality being judged by number of connections with diverse areas of math): -The foundations of analysis being tightened up by people like Cauchy / Weierstrauss / etc. After this...

In car, will reply properly when I get home :). There definitely isn't a single subject you can pinpoint above all others though.

It was in the 1800s, how was it one of the first? It was significant in the subject of polynomial equations, and in this specific subject the cubic formula was the first of a sequence of dominoes after a long period of stagnant theory. The theory of polynomial equations is far from being central...

This proof would need to be pretty long to make any sense to HSC students, or even early undergrads. A decent amount of abstract algebra needs to be developed for this proof. In what sense do you think the Abel-Ruffini theorem jump-started modern mathematics? Galois theory is beautiful but it...

Classic one. For arbitrary polynomial equations of degree 1,2,3,4 over the complex numbers, we can find the solutions in terms of the coefficients, the basic operations of arithmetic, and radicals (taking roots of some degree). For degree 5 and greater, one can prove that this is not generally...

Yes, I know of these experiments. I just haven't learned the physics well enough to understand the connection with the zeta function. (So I definitely cannot give a satisfactory answer to your question without copying/pasting someone elses response lol.) (Tbh, this sort of thing has always...

Some of these statements are pretty fun to prove and not too difficult btw. People should post them in the undergrad marathon!

With all of these zeta function things, people often get confused by what is meant by these divergent sums having values. The function \zeta(s) is defined by the Dirichlet series \sum_n n^{-s} only where it converges, which is the half-plane where the real part of s exceeds 1. Elsewhere in the...

Even if you don't accept the axiom of choice (which is a bit limiting, but some minority of mathematicians don't), you would not be able to prove that such a reassembling of the pea into the sun is impossible. (Because the axiom of choice is consistent with the other axioms of set theory.) This...

If you have a small ball in 3 dimensional space, it is possible to decompose it as a union of a finite number of sets, which can be moved by rotations and translations such that the pieces never overlap and such that the final object constructed is an arbitrarily large ball. Colloquially, one...

So a simple counterexample to the attempt at the differentiation under the integral question is given by: f(x,t)=x, a(x)=a, b(x)=b. You might find it easier to split the problem up into the two individual things going on here a) the interval of integration is changing as x changes b) the...

I am reasonably sure that your expression for (1) is not correct (although it looks close to what one of the terms in a correct expression is, it might even match this term), I can provide a counter-example after doing some housework if you would like. Arbitrary linear ODE with constant...

^ This result is the rigorous reasoning that solves pretty much all of the problems you have posted today, so make sure you understand it.