Top ten open problems in Number Theory

We look at 10 popular open problems in number theory:

1) Perfect number problem
2) Landau's problem
3) Goldbach problem
4) Prime twin problem
5) Riemann hypothesis
6) abc conjecture
7) Beal problem
8) Legendre problem
9) Collatz problem
10) Euler brick problem

[hepmiau]

great video! A good thing about analytical/combinatorial number theory is that it’s relatively easier to explain. As someone studying algebraic number theory/arithmetic geometry, I always have a hard time explaining when my family or non-math friends ask what I’m working on. Calegari’s ICM video this year seems a great introduction, but even then it requires some university level math to understand!

[gkatoch]

Very informative, thanks. I'd like to point out very minor typo that at 1:53 the images of Hardy and Littlewood appear to be swapped.

[adandap]

Excellent video, thank you. At the risk of doing "engineering number theory", it seems suggestive that the gap between (n+1)^2 and n^2 goes like 2n, given Betrand's result with a gap of length n. Seems like there would be 'twice as much chance' of finding a prime. Clearly there is a lot more to it!

[rosiefay]

0:18 Your clue could also relate to the search for Mersenne primes, and to the conjecture that there are infinitely many Mersenne primes.

[erichodge]

Also known as "The Top Ten Career-Killers In Mathematics".

[BritishBeachcomber]

Collatz is one of my favourites because it's so simple to describe, so difficult to prove.

[中井誠二]

Too many problems are about primes. Dividing into two videos, one about primes, and the other not about primes, may makes them easier to understand.

[nzuckman]

Ngl I jumped when Legendre suddenly came to life lmao

[volkerl.]

"Are there infinitely many primes of the form n^2+1" has a completely different meaning than "Are THE infinitely many primes of the form n^2+1".

[toddbeamer]

A nice presentation, even for someone who is not a mathematician. I wonder what you think of the work of Shinichi Mochizuki. It would be interesting to get an objective view of how his work is regarded by other number theorists.

[erockbrox]

I'm a mathematician, and let me tell you why prime numbers are so hard to use to prove things.

It is because prime numbers are probabilistic. This means they are not well defined. The only exists based on a probability.

This is why you see people using approximation equations and such to describe them. Its hard to describe them individually, but as a whole, you can do an approximation function.

[MichaelRothwell]

Very nice summary. Problem 2) Landau's problem seems to be missing in your list in the description.

[nahidhkurdi]

The name labels of Hardy and Littlewood are flipped. Correct the mistake, please.

[rubenscabral]

Continues also has the number (3) number (7) that controls the primes to infinity

[rubenscabral]

I will talk about the riemann hypothesis the (0) is because of the (12) limit of the odd or primes (3) and (7) is valid up to (120) from 3 to 97 is valid up to (10.201)

[Lolwutdesu]

Wow, how did u get the images to move? 😮

[anubhavchauhan]

In the goldbach conjecture, what about 4?
Since the sum must be formed bh two distinct primes, and the primes below 4, i.e., 2 and 3, do not add to form 4.

[tokajileo]

as far as I know there is a proof that there must be a prime between 2 numbers distant 600 or something. (as a result to find proof for the twin prime conjecture). If it is true then the legendre conjecture can be tested up to a finite value and above that it comes from the mentioned proof you just have to test to the number n where (n+1)^2-n^2=proven prime gap ?

[dewaard]

I wonder if we'll ever see a proof of any of these that will become widely accepted.

[HIVEEX]

The collatz conjuncture wouldn't make sense if 1 wasn't even nor odd
1 isn't a prime number tho can we prove that it's neither an odd number ? Or is it technically impossible since we all know that 1 is an odd number