507 – Problem list (I)

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?

Part II.