In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple $\left(x\right)={gx+h_j}_{j=1}^k$ of linear forms in ℤ[x], the set $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that $\left(n\right)={gn+h_j}_{j=1}^k$ contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps ${\delta }_1,...,{\delta }_m$ form an increasing (resp....