I am currently trying to find some good exercises in analytic number theory, suitable for undergraduates.
I have mentioned the Green-Tao theorem for arithmetic progressions of primes but I am struggling to find any interesting applications/problems.
Does anyone know of any?
Exercise/Question: Is the Green-Tao theorem also true for composite numbers, i.e., are there arithmetic progressions $an+b$ with $gcd(a,b)=1$ of arbitrarily large length consisting only of composite numbers ? For example, the progression $7n+1$ gives three composite numbers $8,15,22$ for $n=1,2,3$.
Hint: A solution can be found at MSE, question $p=164513$ prime.
Edit: A problem which uses the Green-Tao theorem could be: Show that there are arbitrarily long arithmetic progressions consisting of numbers which are the sum of two squares.