17-year-old proves new theorem related to Carmichael numbers

[joejoefla]

https://www.youtube.com/watch?v=Kqi_6v2RGB0

[Uncwilly]

Please see:
https://www.mersenneforum.org/showth...postcount=3026

[joejoefla]

Originally Posted by Uncwilly

Please see:
https://www.mersenneforum.org/showth...postcount=3026

Sorry!

[ixfd64]

https://quantamagazine.org/teenager-...likes-20221013

[Uncwilly]

Please see:
https://www.mersenneforum.org/showth...640#post615640

[sweety439]

Originally Posted by ixfd64

https://quantamagazine.org/teenager-...likes-20221013

He proved that for every enough large number n, there is at least one Carmichael number between n and 2*n? (I think that n=3301 is enough large)

Like Chebyshev's theorem: For every n>1, there is at least one prime between n and 2*n exclusive (I also have a conjecture: For every n>1, there is at least one twin prime pair between n and 3*n exclusive)

[LaurV]

Originally Posted by sweety439

(I also have a conjecture: For every n>1, there is at least one twin prime pair between n and 3*n exclusive)

You have first to prove you have enough twin pairs. Maybe there is only a limited (i.e. finite) number of them?
(at least for Carmichael numbers, we knew there are enough of them...)

[R. Gerbicz]

Originally Posted by LaurV

You have first to prove you have enough twin pairs. Maybe there is only a limited (i.e. finite) number of them?

False, but that is trivially true that if there is a twin prime in [n,2n] then there are infinitely many.
And notice that it is even the case for the proof of Chebyshev theorem from Paul Erdos. To give a much weaker theorem:
there is a prime between m and 2^m, for this use the Fermat number F(n)=2^(2^n)+1 for n=ceil(log2(m))-2. (the rest is homework).
And for this we are not using that there are infinitely many primes, but reaching a "Chebyshev type" result, and from that follows that there are infinitely many primes.