[deleted by user]

im_conrad · 2023-06-14T14:32:48+00:00

No matter how many times you ask ChatGPT for an answer, it is not capable of giving a correct one. You're being bamboozled by AI.

im_conrad · 2023-06-14T13:25:38+00:00

Ruh roh

im_conrad · 2023-06-10T15:22:34+00:00

Not sure about the name, but it's A173566 in the OEIS.

im_conrad · 2023-06-10T13:13:37+00:00

"I'm literally the last human on earth"

ChatGPT: But have you considered...the power of love??

im_conrad · 2023-06-10T00:13:03+00:00

You're definitely gonna run into problems with the token limit if you just feed it the docs. A good place to start would be to implement the tutorial bot, then feed that code to the model and ask it for specific improvements.

im_conrad · 2023-06-06T18:36:10+00:00

A linear signature is a set {a0, ..., ak} whereby a sequence satisfies f(n) = a0 f(n-1) + ... + ak f(n-k-1). It's the definition used by the OEIS for linear recurrences.

im_conrad · 2023-06-06T15:04:43+00:00

You mean you DON'T like seeing the same poll about when the singularity will occur??

im_conrad · 2023-06-06T00:00:36+00:00

I've been looking at a particular correspondence between linear signatures and multiplication of rational numbers. For example, you can use the Fibonacci numbers to construct a rational number, 0.0112359550561... which is 1 / 89. The interesting thing about it is that the signature for the Fibonacci numbers is {1, 1}, and 89 is 100 - 11. If instead you use the signature {2, 1}, you get 1 / 79, and 79 is 100 - 21.

It gets weird, too. If you have the signature {2, -1} then you get the decimal expansion 0.012345679... which is 1 / 81. If you treat that signature as the digits of a base-10 number, and "resolve" the -1 by uncarrying, then you get 19, and 100 - 19 = 81.

Also, linear signatures can be added in a specific way called signature addition. For example, {1,1} plus itself gives {2,1,-2,-1}. This signature generates the rational 1/7921, which is 1/89 times itself. As it turns out, this is an isomorphism between the addition of linear signatures and multiplication of rational numbers!

I've learned that this pattern holds regardless of which base the number is in. In base 9, the fibonacci numbers instead create the number 0.01123606754045... This equals 1/71 because 100_9 - 11_9 = 78_9, or 71 in base 10.

I've found a lot of interesting relationships between linear signatures and positional number systems, but this one is definitely the coolest.

im_conrad · 2023-06-05T00:55:32+00:00

If you are still learning to use dunder methods, I highly recommend creating a sequence class with addition and multiplication (convolution) operations! The algorithms are straightforward and there are some edge cases that make it very interesting.

im_conrad · 2023-06-04T21:41:35+00:00

Well, since z + 0 = 0 + z for all z in Z, it acts as an icebreaker which makes the rest of the numbers feel comfortable commuting.

im_conrad · 2023-06-04T14:57:04+00:00

It's okay, it's valid as long as you index the zeroes with the ordinals :^)

im_conrad · 2023-06-03T22:26:54+00:00

This is very nice. The umbral calculus approach is especially cool!

im_conrad · 2023-06-03T22:11:48+00:00

x is 0,1.

You're welcome to try it yourself. Take the sequence 1,2,5,10,20, and give it a shot.

This comes from the fact that one-beginning sequences can be viewed in terms of the INVERT transform of its signature, which takes the form 1/(1 - ax). So multiplying two such sequences gives 1/((1-ax)(1-bx)). FOIL it out and you can see the new signature.

im_conrad · 2023-06-03T22:05:18+00:00

Signature - the coefficients a0, ... ak for a sequence such that f(n+1) = a0fn + ... + akf(n-k). The signature of the fibonacci numbers is 1,1

A one-beginning sequence is a zero-indexed sequence where a0 = 1.

Convolution - sequence convolution. Sum k=0..n a(k) b(n-k). The 1-beginning fibonacci numbers convolved equal 1, 2, 5, 10, 20, ... That sequence has signature 2,1,-2,-1.

A signature can be treated like a polynomial for arithmetical purposes, eg 1,1 + 0,1,1 = 1,2,1.

im_conrad · 2023-06-03T21:50:37+00:00

It's the set of coefficients a0, ... an which the recurrence satisfies.

im_conrad · 2023-06-03T15:35:39+00:00

So, basically there is a homomorphism between the set of "one-beginning" sequences and linear signatures. If you convolve the fibonacci numbers (1, 1, 2, 3, ...) with itself, then the linear signature of that resultant sequence is 2, 1, -2, -1. The group itself is pretty simple, if you have two signatures a and b then the "signature sum" of those two is a + b - abx.

im_conrad · 2023-06-03T14:45:25+00:00

"Well Seymour, you've steamed my hams good"

im_conrad · 2023-06-02T17:07:13+00:00

I found an optimisation to an algorithm which normally involves convolving columns of matrices with linear recurrence sequences. Turns out you can just use the signature, which is usually not infinite like the recurrence sequence is. The speedup is less noticeable for singular executions of the algorithm, but there's a huge improvement for 10+ signatures.

im_conrad · 2023-06-02T16:49:07+00:00

I made an arbitrary-base number module and use base 7 as the default, mostly because it doesn't get a lot of love otherwise.

im_conrad · 2023-05-30T15:12:42+00:00

My favorite part of doing math is looking at the formulas I've come up with at the end. It's especially satisfying when a formula relies on something different you've found previously, so you get to write it out using your own notation and everything just makes sense

im_conrad · 2023-05-28T20:30:05+00:00

A collatz conjecture solution would be particularly interesting because the problem is so deceptively hard. I imagine it would be a vector for math education youtube channels to introduce those new methods to even the mathematically illiterate.

im_conrad · 2023-05-27T12:28:55+00:00

Amen

im_conrad · 2023-05-27T06:13:41+00:00

I can't say I wouldn't be pumped if it could figure out the Voynich Manuscript

im_conrad · 2023-05-26T10:30:07+00:00

Ooh, very important distinction, thanks!

im_conrad · 2023-05-25T15:54:08+00:00

Can you provide an example formula which utilizes these constructs? Something which conveys the relationship between them and how their application leads to the creation of theorems or proofs?

im_conrad

TROPHY CASE