1. ## Re: Interesting mathematical statements

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 interested me less than mathematics, but I might read a bit in the near future to see if I can give a good answer.)

2. ## Re: Interesting mathematical statements

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 possible!

3. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 possible!
Lol proof that there is no 'quintic formula' would interest people.

4. ## Re: Interesting mathematical statements

Another classic that any E4 MX2 student knows:

$\frac {\pi}{4} = 1 - \frac {1}{3} + \frac {1}{5} - \frac {1}{7} + \dots$

5. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 possible!
Abel–Ruffini theorem. Yeah, this was one of the things that jump started the development of modern mathematics. If this had never been proven, we wouldn't have computers, since the advent of modern logic, information theory and computational mathematics arose from the mathematical revolution of the 1900s.

But I digress.

$\sum_{n=1}^{\infty} \tan^{-1} \frac{1}{F_{2n+1}}=\frac{\pi}{4}$

$Where F_m is the \mathrm {m}^{\mathrm {th}} Fibonacci number.$

6. ## Re: Interesting mathematical statements

Classics and more classics:

Typically, the amount of petals on a plant is a Fibonacci number due to it's relationship with the golden mean.

7. ## Re: Interesting mathematical statements

Originally Posted by leehuan
Classics and more classics:

Typically, the amount of petals on a plant is a Fibonacci number due to it's relationship with the golden mean.
Extending on this fact, the petals are typically arranged so that each new petal is separated from the previous one by the golden angle.

8. ## Re: Interesting mathematical statements

Originally Posted by leehuan
Lol proof that there is no 'quintic formula' would interest people.
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.

Abel–Ruffini theorem. Yeah, this was one of the things that jump started the development of modern mathematics. If this had never been proven, we wouldn't have computers, since the advent of modern logic, information theory and computational mathematics arose from the mathematical revolution of the 1900s.

But I digress.

$\sum_{n=1}^{\infty} \tan^{-1} \frac{1}{F_{2n+1}}=\frac{\pi}{4}$

$Where F_m is the \mathrm {m}^{\mathrm {th}} Fibonacci number.$
In what sense do you think the Abel-Ruffini theorem jump-started modern mathematics? Galois theory is beautiful but it isn't nearly as central as you make it out to be here. (Also, what does it have to do with what was done in the early 1900s and the mathematics that led to the invention of computing?)

9. ## Re: Interesting mathematical statements

Another cliche, but possibly interesting to the MX2 student.

$\int _{ -\infty }^{ \infty }{ { e }^{ -{ x }^{ 2 } }dx } =\sqrt { \pi }$

10. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 isn't nearly as central as you make it out to be here. (Also, what does it have to do with what was done in the early 1900s and the mathematics that led to the invention of computing?)
Everything was setup perfectly, the Abel Ruffini theorem was one of the first dominoes that toppled over in mathematical progress. I believe the first domino to fall over was the general solution to cubic polynomial equations. Well perhaps my remark on logic was a bit unfounded, but Information theory was worked on around the same time.

11. ## Re: Interesting mathematical statements

Everything was setup perfectly, the Abel Ruffini theorem was one of the first dominoes that toppled over in mathematical progress. I believe the first domino to fall over was the general solution to cubic polynomial equations. Well perhaps my remark on logic was a bit unfounded, but Information theory was worked on around the same time.
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 to mathematics though, and has very little to do with logic/information theory.

12. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 to mathematics though, and has very little to do with logic/information theory.
What kinds of things are considered central to mathematics? The theory of real numbers?

13. ## Re: Interesting mathematical statements

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

14. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
What kinds of things are considered central to mathematics? The theory of real numbers?
Everything is connected...
It's like an infinitely complicated web that gets more and more complicated as time passes on... So it's likened to brains and the internet and human society in general...
The Riemann Hypothesis has a massive number of connections to areas in modern mathematics, as does the ABC conjecture. I'd provide more detail, but these things are so connected to virtually everything that I struggle to get across the immensity of this network.

15. ## Re: Interesting mathematical statements

Originally Posted by InteGrand
What kinds of things are considered central to mathematics? The theory of real numbers?
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 we were able to do analysis in far more general settings, and actually be sure of our conclusions.

-The work on the foundations of mathematics in the early 20th century, including Godel's results. They might have dealt a crippling blow to our ambitions of having a completely satisfactory foundation for mathematics, but at least it led to a greater understanding of how formal systems work.

-Calculus as originally developed by Newton/Leibniz/etc. It might not have been entirely rigorous at the time, but the physical applicability was immediately obvious.

-Point-set topology developed in the 20th century (in its current form), this is super important to many fields.

-The rigorous development of abstract algebra at the end of the 19th century / start of the 20th. This was well after the work of people like Galois/Abel on polynomial equations, and is considerably more general / abstract in its outlook.

-The development of differential geometry and more recently algebraic geometry, which are very different to the classical subject of geometry studied millenia ago.

These are kind of the roots of the core "branches" of modern mathematics. There are other smaller areas like number theory and information theory of course.

If you view mathematics as like a tree, then the listed developments are some of the big thick branches at the bottom, near the foundational trunk. Galois theory is some small offshoots from the algebra branch, that also intersects with some other things like the number theoretic part of the tree. It is harder to classify things like coding and information theory in terms of those core branches, but they are certainly more minor in terms of how much mathematics is related to / depends on them.

16. ## Re: Interesting mathematical statements

The Riemann Hypothesis has a massive number of connections to areas in modern mathematics, as does the ABC conjecture. I'd provide more detail, but these things are so connected to virtually everything that I struggle to get across the immensity of this network.
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 interconnectedness. Things like the Langlands program give a better taste of it imo. Or Wiles work on FLT.)

17. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 interconnectedness. Things like the Langlands program give a better taste of it imo. Or Wiles work on FLT.)
Fermat's last Theorem built a bridge between two seemingly unrelated areas of mathematics. I doubt that is a good stand-alone example as well. The Langlands program shakes off a similar vibe.

18. ## Re: Interesting mathematical statements

Fermat's last Theorem built a bridge between two seemingly unrelated areas of mathematics. I doubt that is a good stand-alone example as well. The Langlands program shakes off a similar vibe.
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 that one example can demonstrate.

19. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
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 that one example can demonstrate.
That is where we split trains of thought. I went for quantity of connectedness, you went for quality of connectedness.

But I know a few qualitative examples.

The surprising "monstrous moonshine" connection between j-invariants and monster groups was a shocker to the mathematical community when the suggestion of relations was published, and took 15 years to prove. This connection between group theory and number theory was unexpected and hit the mathematical community by surprise.

Another link for j-invariants is with algebraic number theory.
$e^{\pi\sqrt{163}} \approx 640320^3 +744$
This absurdly strange approximation is not a coincidence, it is a result that follows from q-expansion of the j-invariant, with 163 being the last Heegner number.

20. ## Re: Interesting mathematical statements

That is where we split trains of thought. I went for quantity of connectedness, you went for quality of connectedness.

But I know a few qualitative examples.

The surprising "monstrous moonshine" connection between j-invariants and monster groups was a shocker to the mathematical community when the suggestion of relations was published, and took 15 years to prove. This connection between group theory and number theory was unexpected and hit the mathematical community by surprise.

Another link for j-invariants is with algebraic number theory.
$e^{\pi\sqrt{163}} \approx 640320^3 +744$
This absurdly strange approximation is not a coincidence, it is a result that follows from q-expansion of the j-invariant, with 163 being the last Heegner number.
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.

21. ## Re: Interesting mathematical statements

Continuing off of what I just talked about, the Heegner numbers are connected to the quadratic polynomial:
$P(n) \equiv n^2 + n + 41$
This polynomial outputs distinct prime numbers for the numbers 0 to 39
$P(n) \equiv n^2 + n + p$
will also output distinct primes for the numbers 0 to p-2, if and only if, the discriminant of 1-4p is equal to the negative of a Heegner number.
Unfortunately, due to the fact that there are only a finite number of Heegner numbers, there are only a finite number of quadratic equations with this beautiful property. Specifically, only 6 of the 9 Heegner numbers have this property.

22. ## Re: Interesting mathematical statements

Continuing off of what I just talked about, the Heegner numbers are connected to the quadratic polynomial:
$P(n) \equiv n^2 + n + 41$
This polynomial outputs distinct prime numbers for the numbers 0 to 39
$P(n) \equiv n^2 + n + p$
will also output distinct primes for the numbers 0 to p-2, if and only if, the discriminant of 1-4p is equal to a Heegner number.
Unfortunately, due to the fact that there are only a finite number of Heegner numbers, there are only a finite number of quadratic equations with this beautiful property. Specifically, only 6 of the 9 Heegner numbers have this property.
Well 1-4p has to be the negative of a Heegner number, not a Heegner number itself, but yes this is pretty cool.

23. ## Re: Interesting mathematical statements

Originally Posted by glittergal96
Well 1-4p has to be the negative of a Heegner number, not a Heegner number itself, but yes this is pretty cool.
Missed and fixed.

Next bizarre truth

$\sum_{n=1}^{\infty}\frac{n^3}{2^n} = 26$

Supposedly first proven by Bernoulli.

Here is the Polylogarithmic evaluation of the infinite sum.

$\mathrm {Li}_s(z) = \sum_{k=1}^\infty \frac{z^k}{k^s}$

$Taking s = -3 and z = \frac{1}{2} yields our desired sum.$

$The particular case of s = -3 has the form: \mathrm {Li}_{-3}(z) = \frac{z(1+4z+z^2)}{(1-z)^4}$

$Plugging in z = \frac{1}{2} evaluates our series to be exactly 26.$

$\noindent Please post Bernoulli's solution on this thread if it exists and you happen to find it.$

24. ## Re: Interesting mathematical statements

paradoxica, r u even year 12

25. ## Re: Interesting mathematical statements

Originally Posted by turntaker
paradoxica, r u even year 12
My legal age is not the same as my mental age, let's go with that.

But also my academic age and my psychological age are different, due to neurodevelopmental abnormalities.

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