The following are four of the most important problems regarding prime numbers:
1. There are infinitely many primes that differ by 2.
2. Every positive even integer is the sum of two primes.
3. There exists at least one prime between two consecutive integer squares.
4. There are infinitely many primes differing by 1 from an integer square. Despite their propularity, none of them has ever been solved.
However, in the last 3 years there has been tremendous progress regarding the first problem,
see this Quanta magazine article.
In this course we shall give a glimpse towards these recent developments.
Namely, we shall prove that there are infinitely many pairs of consecutive primes that differ at most by 600.
The only prerequisites for this course is a basic course in Number Theory.
The exercises will count for 30% of the final grade.
There will be a project that will contribute 30% towards the final grade.
The remaining 40% will be given based on the final oral exam.
Notes will be provided in this webpage during the course.