---

Statement

Setting (pp. 19--21). In Cramér's model, quoted by Granville in Cramér's own words and introduced on p. 19 as Cramér's work of 1937, there is an urn $U_n$ for each $n$, and a white ball is drawn from $U_n$ with probability $1/\log n$ for $n>2$, independently; the numbers $P_1<P_2<\cdots$ of the urns that give white balls form a random increasing sequence. Cramér shows that with probability $1$,

$$\limsup_{n\to\infty}\frac{P_{n+1}-P_n}{(\log P_n)^2}=1,$$

and takes this as a suggestion that a similar relation may hold for the primes $p_n$.

Conjecture (p. 21, (14)). Granville reads this as the prediction that the largest gap between consecutive primes up to $x$ is about $\log^2x$:

$$\max_{p_n\le x}\,(p_{n+1}-p_n)\sim\log^2x.\qquad(14)$$

He records that (14), "or the weaker $O(\log^2x)$", is known as "Cramér's Conjecture" (p. 21, quoted).

Shanks's form (p. 21). Granville reports that Shanks reformulated (14) to suggest that the first gap between consecutive primes of size $>g$ occurs with $p_n=e^{\{1+o(1)\}\sqrt g}$.

Record gaps (pp. 21--22). A table on p. 22 lists the record-breaking gaps up to $10^{14}$, from $p_n=31397$ (gap $72$, ratio $(p_{n+1}-p_n)/\log^2p_n=0.6715$) to $p_n=19581334192423$ (gap $778$, ratio $0.8177$). Granville says on p. 21 that these computations indicate that Cramér may well have been right.

The paper proves nothing about (14). On p. 24 it argues that a sieve-corrected model contradicts (14); see the heuristic on p. 24.

Source. A. Granville, Harald Cramér and the distribution of prime numbers, Scand. Actuar. J. 1995, no. 1, 12--28: pp. 19--22. The edition read is identified on the source card.

Read depth. Claims checked: the model, (14), Shanks's form and the table were read clause by clause on the printed pages. Nothing here is independently reviewed.

Proof pointer

None: (14) is a conjecture. Cramér's probabilistic theorem for the model is quoted in Cramér's words, and Cramér proves it; the survey only reports that Cramér shows the event $E_m$ has probability of order $m^{-c}$ and applies Cantelli's lemma.

Dependencies

Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23--46, listed in the survey's references (p. 28); the survey attaches no citation to the quotation itself.

Bears on

• Problem 680: the paper does not mention the problem or the least prime factor of $n+k$. As an observation of this page, the threshold $e^{(1+\epsilon)\sqrt k}$ in the problem's second question has the shape of Shanks's form of (14), where a gap of size $g$ is first expected near $e^{(1+o(1))\sqrt g}$. Nothing in the paper proves or disproves either question of the problem.