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:
1=1
3=1+2
7=1+2+2**2
..
And so utilises a bit of syntactical sugar that I think can be a bit confusing where he calculates for the viewer only 2**0 and 2**1 and requires the viewer to do rest of the power calculations and the sums in their head to see the relation.
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
This has an interesting characteristic that an infinite sum in the reals that diverges, "gets larger", converges, "settles on a number".
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!
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.
Grant shows:
And so utilises a bit of syntactical sugar that I think can be a bit confusing where he calculates for the viewer only 2**0 and 2**1 and requires the viewer to do rest of the power calculations and the sums in their head to see the relation.
This has an interesting characteristic that an infinite sum in the reals that diverges, "gets larger", converges, "settles on a number".
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...