Figures & data: Gaps Between Consecutive Primes and the Exponential Distribution

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Fig. 1 Moments $μ_{k, n}^{'} (k = 1, 2, 3, 4)$ of the first n gaps between consecutive primes (solid dots) from k = 1 (bottom row) to k = 4 (top row); and corresponding kth moments $(k = 1, 2, 3, 4)$ of exponential distributions $Exp (1 / log n)$ (solid lines). The dots and the lines are calculated independently with no adjustment of parameters.

Table 1 For each upper limit $x = 2^{t}$ , this table shows the exponent t of 2, the number n of gaps between consecutive primes (not counting the odd first gap), the moments $μ_{k, n}^{'} (k = 1, 2, 3, 4)$ , and the maximal gap G_n .

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Fig. 2 Maximal gap G_n (solid blue dots) among the first n gaps and four conjectured models: ${(log n)}^{2}$ (solid black line); ${(log p_{n})}^{2}$ (solid black line with + markers); $2 e^{- γ} {(log n)}^{2}$ (dashed purple line), where $2 e^{- γ} \approx 1.1229$ ; and $2 e^{- γ} {(log p_{n})}^{2}$ (dashed purple line with + markers). The dots and lines are calculated independently with no adjustment of parameters. While the great majority of the values of G_n are better approximated by ${(log n)}^{2}$ (solid black line) than by the other models, the seven exceptional values of n with $G_{n} > 2 e^{- γ} {(log n)}^{2}$ in suggest that $\underset{n \to \infty}{lim sup} G_{n} / [2 e^{- γ} {(log n)}^{2}]$ may exceed 1.

Table 2 Seven maximal gaps G_n that exceed ${(log n)}^{2}$ and $2 e^{- γ} {(log n)}^{2}$ , the index n of the prime p_n that begins the maximal gap, the prime p_n that begins the maximal gap, ${(log p_{n})}^{2}$ , and $2 e^{- γ} {(log p_{n})}^{2}$ .

Display Table

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Gaps Between Consecutive Primes and the Exponential Distribution

Table 1 For each upper limit $x = 2^{t}$ , this table shows the exponent t of 2, the number n of gaps between consecutive primes (not counting the odd first gap), the moments $μ_{k, n}^{'} (k = 1, 2, 3, 4)$ , and the maximal gap G_n .

Table 2 Seven maximal gaps G_n that exceed ${(log n)}^{2}$ and $2 e^{- γ} {(log n)}^{2}$ , the index n of the prime p_n that begins the maximal gap, the prime p_n that begins the maximal gap, ${(log p_{n})}^{2}$ , and $2 e^{- γ} {(log p_{n})}^{2}$ .

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Gaps Between Consecutive Primes and the Exponential Distribution

Figures & data

Table 1 For each upper limit x=2t, this table shows the exponent t of 2, the number n of gaps between consecutive primes (not counting the odd first gap), the moments μk,n′(k=1,2,3,4), and the maximal gap Gn .

Table 2 Seven maximal gaps Gn that exceed ( log n)2 and 2e−γ( log n)2, the index n of the prime pn that begins the maximal gap, the prime pn that begins the maximal gap, ( log pn)2, and 2e−γ( log pn)2.

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Table 1 For each upper limit $x = 2^{t}$ , this table shows the exponent t of 2, the number n of gaps between consecutive primes (not counting the odd first gap), the moments $μ_{k, n}^{'} (k = 1, 2, 3, 4)$ , and the maximal gap G_n .

Table 2 Seven maximal gaps G_n that exceed ${(log n)}^{2}$ and $2 e^{- γ} {(log n)}^{2}$ , the index n of the prime p_n that begins the maximal gap, the prime p_n that begins the maximal gap, ${(log p_{n})}^{2}$ , and $2 e^{- γ} {(log p_{n})}^{2}$ .