This is the list of “problems of the day” mentioned through the course.
(Thanks Nick Davidson and Summer Hansen.)
- Frankl’s union-closed sets problem: If a finite collection of finite non-empty sets is closed under unions, must there be an element that belongs to at least half of the members of the collection?
- The inverse Galois problem: Is every finite group a Galois group over ?
- 1. Are there infinitely many Mersenne primes? 2. Are there infinitely many Fermat primes?
- For every positive , is there a prime between and ?
- Does the dual Schroeder-Bernstein theorem imply the axiom of choice?
- The Schinzel–Sierpiński conjecture: Is every positive rational of the form for some primes and ? (The links require a BSU account to access MathSciNet.)
- Are there infinitely many twin primes?
- Are there any odd perfect numbers?
- Is the Euler-Mascheroni constant irrational?
- Is a primitive root modulo for infinitely many primes ? More generally, does Artin’s conjecture hold?