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?

Dr. Caicedo,

The inverse Galois problem (30 August) was covered before the Mersenne prime problem (1 September).

Fixed. Thanks!

Dr. Caicedo,

My notes also show that between the inverse Galois problem and the prime between n^2 and (n+1)^2 problem, you discussed whether or not there are infinitely many Fermat primes (also on 1 Sept).

Also, you discussed odd perfect numbers on 15 Sept, the class before the Euler-Mascheroni constant problem. Everything else coincides 🙂

Hehe. Fixed. Thanks!

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