Let q > r ≥ 1 be coprime integers. Let P c = P c (q, r, H) be an increasing sequence of primes p satisfying two conditions: (i) p ≡ r (mod q) and (ii) p starts a prime k-tuple with a given pattern H.

If p k is the kth prime, the Firoozbakht conjecture states that the sequence (p k ) 1/k is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify the Firoozbakht conjecture for primes up to four quintillion (4 × 10 18 ). Mathematics Subject Classification: 11N05

In the framework of Cramér's probabilistic model of primes, we explore the exact and asymptotic distributions of maximal prime gaps. We show that the Gumbel extreme value distribution exp(− exp(−x)) is the limit law for maximal gaps between Cramér's random "primes." The result can be derived from a general theorem about intervals between discrete random events occurring with slowly varying probability monotonically decreasing to zero. A straightforward generalization extends the Gumbel limit law to maximal gaps between prime constellations in Cramér's model.

Let q > r ≥ 1 be coprime positive integers. We empirically study the maximal gaps G q,r (x) between primes p = qn + r ≤ x, n ∈ N. Extensive computations suggest that almost always G q,r (x) < ϕ(q) log 2 x. More precisely, the vast majority of maximal gaps are near a trend curve T predicted using a generalization of Wolf's conjecture:where b = b(q, x) = O q (1). The distribution of properly rescaled maximal gaps G q,r (x) is close to the Gumbel extreme value distribution. However, the question whether there exists a limiting distribution of G q,r (x) is open. We discuss possible generalizations of Cramér's, Shanks, and Firoozbakht's conjectures to primes in residue classes.

Let q > r ≥ 1 be coprime integers. Let R(n, q, r) be the nth record gap between primes in the arithmetic progression r, r + q, r + 2q, . . . , and denote by N q,r (x) the number of such records observed below x. For x → ∞, we heuristically argue that if the limit of N q,r (x)/ log x exists, then the limit is 2. We also conjecture that R(n, q, r) = O q (n 2 ). Numerical evidence supports the conjectural a.s. upper bound R(n, q, r) < ϕ(q)n 2 + (n + 2)q log 2 q.The median (over r) of R(n, q, r) grows like a quadratic function of n; so do the mean and quartile points of R(n, q, r). For fixed values of q ≥ 200 and n ≈ 10, the distribution of R(n, q, r) is skewed to the right and close to both Gumbel and lognormal distributions; however, the skewness appears to slowly decrease as n increases. The existence of a limiting distribution of R(n, q, r) is an open question.Mathematics Subject Classification: 11N05, 11N13, 11N56