Tristan Freiberg: On consecutive primes in tuples

An old conjecture of Chowla, now a theorem of D. Shiu, asserts that, given an arithmetic progression $a \bmod q$ with $(a,q) = 1$, there are infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ such that $p_n = p_{n+1} = a \bmod q$. Assuming the Elliott-Halberstam conjecture and using the work of Goldston-Pintz-Y{\i}ld{\i}r{\i}m on gaps between primes, we will show that we infinitely often have $p_n = p_{n+1} = a \bmod q$ with $p_{n+1} - p_n \ll q^L$ for some absolute constant $L$.