
The Ramanujan Machine: Using algorithms to discover new mathematics - mci
http://www.ramanujanmachine.com/
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thom
Seems slightly less rich than Doug Lenat's Automated Mathematician (which in
turn led to Eurisko which was one of the earliest genuinely interesting AI
systems):

[https://en.wikipedia.org/wiki/Automated_Mathematician](https://en.wikipedia.org/wiki/Automated_Mathematician)

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beagle3
The story of Eurisko doesn't add up. So advanced for its time, not replicated
for 30 years (or at all) and Lenat lost the source code and was not interested
in doing anything similar again, with the next version (Cyc, IIRC) not getting
anywhere close?

I would guess there was a lot more manual handholding that was not documented.

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thom
Yes, I mostly agree here. But even with that manual handholding, it's no
different from current, mostly-supervised AI approaches. Even if you think of
Eurisko as a very clever optimiser when posed problems in a certain way, it
clearly delivered some interesting results. Also, you _can_ find the source
code for large parts of Eurisko if you look hard enough, I believe.

Cyc never seemed in the least bit interesting to me, tbh. Even today several
"we taught our AI common sense!" articles have hit HN, and it's still not
_really_ true.

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Grue3
Well, the continuous fraction for e is pretty well known. Doubt they
discovered anything new here that can't be obtained from the original formula.
On the other hand, the continuous fraction for pi is irregular, so it's
interesting to see what they discovered... but I can't really find any pattern
in the "conjectures" for pi. Take the first one:

pi/−4 = 1/(−1 + 1/(−4 + −2 /(−7 + −9/(−10 + −20/(−13+...))))

What exactly am supposed to prove here? The denominators are an arithmetic
progression but numerators (1, 1, -2, -9, -20, ...) are just some bizarre
sequence without an obvious pattern. The thing with continous fractions is
that every number has one, so the fact that pi is presented as continous
fraction is not impressive in itself.

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gjm11
All their conjectures have low-degree polynomial formulae for both numerators
and denominators, with exceptions allowed in the first couple. So I guess the
nth numerator, counting from 0, is -n(2n-3) which goes 0, 1, -2, -9, -20, ...,
so n=0 is a special case.

It would seem more natural to rewrite it like this

-4/pi = −1 + 1/(−4 + −2 /(−7 + −9/(−10 + −20/(−13+...))))

which avoids the gratuitously different first numerator -- but I guess they
wanted to reproduce results exactly as they happened to emerge from their
program.

I think this is, further, equivalent to the following which avoids some
gratuitous-looking minus signs:

4/pi = 1 + 1/(4 + −2 /(7 + −9/(10 + −20/(13+...))))

Continuing the fraction using the quadratic polynomial I gave above does
indeed seem to make it converge to 4/pi, though not very quickly.

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gjm11
I wonder whether this one might be equivalent to a known infinite _product_
for pi or pi/4 or something. The ratios between successive convergents are all
of the form (a_n+k_n)/a_n where k_n is 1, 1, 3, 3, 15, 15, 105, 105, 315, 315,
... -- i.e., the least common multiples of odd numbers up to n. [EDITED to
add:] This should make you think of the Wallis product formula for pi/4,
though this doesn't seem to be the Wallis product in disguise.

Clearly these k_n divide n!, so maybe the right way to say this is that the
conjecture seems to be equivalent to 4/pi = product (a_n+n!)/a_n where (a_n) =
(4,130,2464,45448,882528,18410640,...) ... though I don't know what that
sequence _is_, haven't shown that it has a nice form, etc. None of (a_n),
(a_n+n!), (a_n+n!/2) seems to occur in OEIS or to be a subsequence of anything
in OEIS.

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nurettin
It seems aptly named because Ramanujan also dealt with infinite series. One of
his most famous equations is the ramanujan summation [1] where he derives that
the sum of infinite series 1 + 2 + 3 + 4 + ... = - 1 / 12

[1]
[https://en.wikipedia.org/wiki/Ramanujan_summation](https://en.wikipedia.org/wiki/Ramanujan_summation)

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amelius
I still don't understand why this derivation is such an achievement.

If anything, it seems like dirty hacking, but this time done by a
mathematician.

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nurettin
I read somewhere that the result is confirmed in applied physics where it
somehow relates to the casimir force, but I'm not a physicist.

~~~
mathandpoop
It's a theorem of analytic number theory. It's just a special value of the
ζ-function. People misunderstand how analytic continuation works and therefore
(wrongly) interpret ζ(-1) as the divergent series 1+2+3+... (The correct
interpretation is as the unique analytic continuation of the holomorphic
function $\sum_{n\geq 1}\frac{1}{n^s}$ defined on the half-plane $\Re(s)>1$ to
$\mathbb{C}\setminus\\{1\\}$.)

It's actually a pretty simple consequence of the functional equation for ζ and
a few special values of Γ. That in turn comes from a theta-function identity
which can be proven using Poisson's summation formula.

What I'm trying to say is that it's legit maths, that has been distorted due
to the shock value of writing the equation "1+2+3+... = -1/12".

If you're looking for a reference, go to Davenport's "Multiplicative Number
Theory". It's short, self-contained, and extremely well-written. Serre's "A
Course in Arithmetic" should also work.

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xvilka
I wonder if it somehow can be integrated with Coq[1] and Univalent
Foundations[2]. Probably these can be used as a more substantial "base" for
this machine.

[1] [https://github.com/coq/coq](https://github.com/coq/coq)

[2] [https://github.com/UniMath](https://github.com/UniMath)

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mometsi
Why on earth did they not call it the Ramanutron??

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thaumasiotes
Possibly because that only really works if you know the correct stress pattern
of "Ramanujan". Americans, at least, are likely to want to stress it on the
"nu".

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bryanrasmussen
Didn't Good Will Hunting teach the proper pronunciation?

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Zenst
I was pondering a few weeks back about the prospects of using ML/AI to find
new ways to factor primes and if their is any sequence in primes and how to
calculate them in a way that you input N and it will produce the Nth prime.

That would be something.

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m3kw9
Can primes be estimated?

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drinfinity
Through my genius I've come to the conclusion that primes are related to
entropy. There is no hope.

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lifthrasiir
Related: Robert Munafo's RIES [1] tries to synthesize increasingly complex
formula which solution is a given number.

[1] [https://mrob.com/pub/ries/](https://mrob.com/pub/ries/)

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mathandpoop
"New mathematics" is a bit of an overstatement. There is a lot more to maths
than continued fractions.

It bugs me a bit that the authors write (I would love for anyone affiliated
with the project to talk to me about this) "Any new conjecture, proof, or
algorithm suggested will be named after you.". No offense, but there are very
few mathematicians out there with that kind of a world view.

Seems a bit like a high school project without proper guidance from a
mathematician.

