4
$\begingroup$

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?

$\endgroup$
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ That is a good question and is aimed at the kind of level I am looking for, but I would like a problem that somehow uses the theorem. $\endgroup$ – fretty Oct 1 '14 at 10:03
  • $\begingroup$ @fretty OK, I have edited the answer. $\endgroup$ – Dietrich Burde Oct 1 '14 at 11:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.