Self-study diary

(This will be updated every day until I'm too tired to study about "impossible" problems.)

This should be $\LaTeX$ compatible, I guess.

19 Dec

Is $1_{\textsf{NP}}$ in $\textsf{NP}$ ? If yes, then $\textsf{NP} = \textsf{co-NP}$ . (and the answer is no)
Formulation of Collatz conjecture as a language.
The language of prime numbers is not in $\textsf{CF}$ . (sad)
Models of computation that are weaker than or equal to finite-state machine and their corresponding powers. (Combinational logic, Sequential logic)
Representation of integers using semiprimes: Can we introduce some better models of integer multiplication algorithm using a blend of prime factorization representation and the usual n-bit representation?
Integer factorization problem: Why is it hard? How could it be sub-exponential but super-polynomial?
What is the running time of Rational sieve?
Latest progress on the Collatz conjecture.
Generalization of the Collatz conjecture: Can we find some patterns and start a new theory to describe a new class of functions related to the Collatz map?

Collatz conjecture implies that the Collatz language is regular so one way of disproving the Collatz conjecture is to use the pumping lemma on the Collatz language and find a contradiction.
Landau's problems: What are they and how are the progresses?

We can unify mathematical proofs with the problem of "deciding whether a statement is true or false" which is a computational problem.
Gödel's incompleteness theorem states the undecidability of math proofs in general case. However, we can still do partial results. This is why proving is hard.
Stated an idea of using AI/machine learning technologies to automatically generate mathematical proofs, since the problem of "verifying proofs" are decidable (easily).
Got an inspiration of looking into the model theory.

Mertens conjecture revisited.
Trying to find an explicit counterexample of the Mertens conjecture.
Riemann hypothesis revisited. Inspired by the Lagarias reformulation in elementary number theory.