Alladi Ramakrishnan Hall

Product of three primes in arithmetic progression

R. Balasubramanian

IMSc

It is a celebrated theorem of Dirichlet that an arithmetic progression a(mod q) contains infinitely many prime numbers provided gcd(a,q)=1. Linnik proved that there exists a constant c>0 such that the least prime in a(mod q) is at most q^c. Even though one expects the result to be true for any c>2 (for large q), the best known result is that it is true for c=5. Now, let us define P_3={natural numbers which can be written as a product of three prime numbers}. In this lecture, we shall discuss the following question: Give a(mod q) with gcd(a,q)=1, what is the bound for the least integer in P_3 which is also in a(mod q).