## 507 – Problem list (I)

September 21, 2010

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 ${\mathbb Q}$?
• 1. Are there infinitely many Mersenne primes? 2. Are there infinitely many Fermat primes?
• For every positive $n$, is there a prime between $n^2$ and $(n+1)^2$?
• Does the dual Schroeder-Bernstein theorem imply the axiom of choice?
• The SchinzelSierpiński conjecture: Is every positive rational of the form $(p+1)/(q+1)$ for some primes $p$ and $q$? (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 $\gamma$ irrational?
• Is $2$ a primitive root modulo $p$ for infinitely many primes $p$? More generally, does Artin’s conjecture hold?