Based on the primes less than 4 × 10 18 , Oliveira e Silva et al. (Math. Comp., 83(288):2033–2060, 2014) conjectured an asymptotic formula for the sum of the kth power of the gaps between consecutive primes less than a large number x. We show that the conjecture of Oliveira e Silva holds if and only if the kth moment of the first n gaps is asymptotic to the kth moment of an exponential distribution with mean log n , though the distribution of gaps is not exponential. Asymptotically exponential moments imply that the gaps asymptotically obey Taylor’s law of fluctuation scaling: variance of the first n gaps ∼ (mean of the first n gaps)2. If the distribution of the first n gaps is asymptotically exponential with mean log n , then the expectation of the largest of the first n gaps is asymptotic to ( log n ) 2 . The largest of the first n gaps is asymptotic to ( log n ) 2 if and only if the Cramér-Shanks conjecture holds. Numerical counts of gaps and the maximal gap Gn among the first n gaps test these results. While most values of Gn are better approximated by ( log n ) 2 than by other models, seven exceptional values of n with G n > 2 e − γ ( log n ) 2 suggest that lim sup n → ∞ G n / [ 2 e − γ ( log n ) 2 ] may exceed 1.