@Sawarnik I'll get the trivial bits first : $f(0) = 0$, $f(-x) = -f(x)$ so this is an odd function.

OK, duplication formula : $f(2x) = 2f(x) + 2x^3$

And perhaps it might help to write down the triplication formula too : $f(3x) = 3f(x) + 8x^3$

The sequence is something like $0, 2, 8, 20, 40, 70, 112$. There is an obvious recurrence, but I will be a bit lazy here and cheat the answer out from OEIS. It's $2 \binom{n}{3}$. So $f(nx) = nf(x) + 2\binom{n}{3}x^3$

@BalarkaSen And then? Ok, you are thinking.

The answer's actually quite simple.

@Sawarnik Did you do it?

@BalarkaSen I had the misfortune of seeing the question and answer at the same time.

@Sawarnik The I will assume it uses some unguessable trick theory.

I.E., It's not easy.

@BalarkaSen Not unguessable at all.

@Sawarnik You saw the answer, so you never know.

=P

My approach is to find some auxilliary function that transforms this into $A(nx) = nA(x)$

I'll throw a stone in the dark : assume $f(x) = A(x) + cx^3$ (inspired by the appearance of the cubic term in the multiplication formulas). Then $f(nx) = A(nx) + cn^3x^3$ and $nf(x) = nA(x) + ncx^3$. Using the multiplication formula, $2\binom{n}{3}x^3 = A(nx) - nA(x) + (n^3-n)cx^3$

Ah, my internet is annoying :(

@BalarkaSen Ok, I ll give you a hint.

No need.

I am done.

Oh.

Not yet, darn.

I think I messed up somewhere above.

Want a hint? But its nice to see you struggle.

@Sawarnik No.

Ah, of course. I only needed a replacement. This is trick.

Well, just get to the point. $f((n+1)x) = A(nx) + cn^3x^3$

This should do.

But then, I am really missing something here.

.

@Sawarnik Now you're just annoying me with those smileys.

Oh, damn. I figured it out.

I didn't really look carefully at the OEIS entry, that's the whole point of the problem. First, it starts with $n = 0$ and first 3 entries are $0$ (obviously) so $f(nx) = nf(x) + 2\binom{n+1}{3}$

Thus, $c(n^3 - n) = (n+1)n(n-1)/3$

Hence $c = 1/3$

And $f(x) = A(x) + x^3/3$, and $A(x + y) = A(x) + A(y)$, continuity guarantees $A(x) = kx$

Done and done.

@BalarkaSen Good work.

:D

@Sawarnik How did they do it?

@BalarkaSen Quite similarly. But notices in the beginning that x^3/3 satisfies it and so ...

Want mor?

@Sawarnik Trick theory is no proof. Note that I have never used tricks, all of it comes rigorously.

@BalarkaSen It is a proof. And its not hard to notice that $x^3/3$.

@Sawarnik It's one like Cleo's posts. No proof unless you know how the trick comes.

Never use trick theory without knowing how it actually comes to play.

@BalarkaSen Doesn't the formula $(x+y)^3=x^3+y^3+3xy(x+y)$ strike you!

@Sawarnik No. I try to formalise more, trick less.

No hell. You do a number theory now in turn, for a change.

:D

Oh no!

Ceasefire.

@Sawarnik Oh yes.

if you complain much I will give you combinatorics.

So do the NT nice and quietly.

@Sawarnik Tell me if you want to do, otherwise I won't bother finding one.

Not sure.

Ok, leave it. I should study vectors and matrices now :D

You should also study other things.

Too much vectors and matrices might lead you to exterior algebras (oof)

@BalarkaSen That's a looong way from now.

@Sawarnik How much have you learned in vectors and matrices?

matrix operations and vector spaces (basis, etc, and blardy blardy blahs)?

@BalarkaSen Yes, that's where I am.

@Sawarnik You just need those two for studying abstract algebra.

In fact, you just need matrix operations right now to study groups.

I'd much prefer you to jump on to abstract algebra.

Grr, this group theory has number theory in it.

@Sawarnik No.

Not at all.

I saw the totient function and mods.

@Sawarnik That's just an application from group theory.

You need no perquisite to NT at all.

Abstract algebra is a pretty self-contained branch of mathematics, if you want to study it. But it has applications to a wide number of topics, if you want to apply it.

Hmm.

Besides do you know that in Romania they teach group theory in 12th?

@Sawarnik No, I never knew that. But it's not odd, undergrad group theory (that's all I know of group theory) is not that big of a deal.

But it's a shame that we are taught group theory first than field theory world wide.

One should first learn field and galois theory, then do groups and whatnots.

:15616445 Yes, because most books don't have such course preferation.

But one should remember that the great mathematicians developed field and galois theory first, not group theory.

An obvious question one would want to ask : Why is abstract algebra called 'algebra'? Can you guess why, @Sawarnik?

:15616480 I asked about why "algebra"

Why not something like "supergeometricalnonsense"

Algebra used to be the things with variables. But hmm, why the abstract and linear and exterior are algebras?

@Sawarnik Yes. Do you want me to answer it?

@BalarkaSen That would have been a better name!

@BalarkaSen Yea.

@Sawarnik The truth's harsh but it's because all of them tries to answer one question : Why is the quintic unsolvable? One might protest that there are loads more question in abstract algebra, more interesting than that. But hey, the whole theory developed through answering that question. It's just because the origin of abstract algebra is galois theory, and it tries to answer the insolvability of quintics, which is a question coming out of algebra (now called theory of equations).

That's scary.

@Sawarnik Star it, if you think so =P

I always try to explain this philosophy of mine to people.

And when they say : What's the big deal about polynomials anyway?

@BalarkaSen Wiki doesn't answer the question too.

I just give them a quote from Lang "You think we know everything about polynomials?"

@Sawarnik It's my personal view of things.

My own mathematical philosophy.

Ah I see.

Yes.

But people often call category theory "abstract nonsense"

@WendiKidd I was promised that all the messages in that discussion would be removed. I'm not comfortable with the fact that they're exposed to the public.

Its quite ironic that the original owners of this room are now banned to be one.

@Sawarnik Excuse me, did you not have the same thought when you removed me from one of the owners?

@ParthKohli Ah, did you not have the same thought when you removed me from one of the owners? Because in that chain, I was the first one to be removed.

@Sawarnik I wonder why you didn't tell rob that I made the room.