Mathematicians Find Long-Sought Building Blocks for Special Polynomials(quantamagazine.org)
quantamagazine.org
Mathematicians Find Long-Sought Building Blocks for Special Polynomials
https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525
5 comments
I'm no mathematician so reading this article was a familiar journey:
- Polynomials and roots of unity, I'm with you.
- L-functions, I've read a little about those.
- P-adic numbers, I've read a little about those.
P-adic L-functions? I'm out.
- Polynomials and roots of unity, I'm with you.
- L-functions, I've read a little about those.
- P-adic numbers, I've read a little about those.
P-adic L-functions? I'm out.
There is something very deep and strange going on in the p-adic L-function domain.
I feel like we're close to something very important that everyone is completely missing.
I feel like we're close to something very important that everyone is completely missing.
What's special about them (what are we missing)?
You add infinite number of positives numbers to each other and you get -1 as a result. p-adic numbers are adding some sense to this seemingly contradictory result. Great intro: https://www.youtube.com/watch?app=desktop&v=XFDM1ip5HdU
Just to give this some more context, since I feel Grant sort of rushes at the end a bit when he covers this and goes from powers of two to 1,3,7,15..
Grant shows:
What is special about it?
In physics you will hear complaints about equations that have infinities that diverge, or "blow up". This can cause many problems in physical models.
This toy example shows one such infinity that blows up in the reals, but converges in the 2-adics.
I think this leaves mathematicians and physicists wondering if this technique can be used to remove, or better understand, the problematic infinities they encounter.
What are we missing?
That's a great question, and one that mathematicians and physicists are trying to answer. Jump in there and give answering it a try! We could always use more help!
https://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B...
Grant shows:
1=1
3=1+2
7=1+2+2**2
..
1+2+4+8+16+..==-1
is the same as saying
2**0+2**1+2**2+2**3+..==-1
If we look at the partial sums, successively adding new powers of two:
power reals 2-adics distance metric
2**0 == 1 == -1
2**0+2**1 == 3 == -1
2**0+2**1+2**2 == 7 == -1
(DIVERGES) (CONVERGES)
and on ad infinitum
What is special about it?
In physics you will hear complaints about equations that have infinities that diverge, or "blow up". This can cause many problems in physical models.
This toy example shows one such infinity that blows up in the reals, but converges in the 2-adics.
I think this leaves mathematicians and physicists wondering if this technique can be used to remove, or better understand, the problematic infinities they encounter.
What are we missing?
That's a great question, and one that mathematicians and physicists are trying to answer. Jump in there and give answering it a try! We could always use more help!
https://en.wikipedia.org/wiki/1_%2B_2_%2B_4_%2B_8_%2B_%E2%8B...
Unsure what specifically the gp was referring to but there is a known relation between L-functions and the prime number sequence, see Riemann hypothesis.
And p-adics are constructed using modulo some prime.
With all these prime relations maybe p-adic L-functions could lend themselves to any number of prime adjacent conjectures: distribution, twins, goldbach, Riemann.
And p-adics are constructed using modulo some prime.
With all these prime relations maybe p-adic L-functions could lend themselves to any number of prime adjacent conjectures: distribution, twins, goldbach, Riemann.
Just to give credit where credit is due: "The code to perform these computations was written by Max Fleischer and Yijia Liu, two undergraduate students of the first author at Duke University." (from the pre-print paper)
Kudos, Max and Yijia!
Very impressive for undergraduates to get involved in 100-year-old unsolved maths problems!
Kudos, Max and Yijia!
Very impressive for undergraduates to get involved in 100-year-old unsolved maths problems!
Fascinating result!
Very happy to see older generations acknowledging code and seeing it as a way to collaborate with younger generations.
Great work!
Quietly linked in the article (uses Sage!): https://github.com/liuyj8526/Computation-of-Elliptic-Units
Very happy to see older generations acknowledging code and seeing it as a way to collaborate with younger generations.
Great work!
Quietly linked in the article (uses Sage!): https://github.com/liuyj8526/Computation-of-Elliptic-Units
Is there anywhere a slightly more in depth article about this?
https://m.youtube.com/watch?v=XFDM1ip5HdU
This article fills in some bits omitted from the video: https://www.quantamagazine.org/how-the-towering-p-adic-numbe...