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Statement

Page numbers are those of the author preprint named on the source card, whose five pages correspond to pp. 207--211 of the journal.

Setting (p. 1). $n\ge0$ and $k\ge1$ are integers; $p_i$ is the $i$-th prime; $\omega(\nu)$ is the number of distinct prime divisors of an integer $\nu>1$, with $\omega(1)=0$; and $N_0=8.5\times10^8$. When $n+1,\ldots,n+k$ are all composite and there are distinct primes $P_i$ with $P_i\mid n+i$ for $1\le i\le k$, Grimm's conjecture is said to hold for $n$ and $k$. Grimm's conjecture (unrestricted) asks for such primes whenever $n+1,\ldots,n+k$ are all composite.

Theorem 1 (p. 1, quoted). "Grimm's Conjecture holds for $n\leq p_{N_0}$ and for all $k$."

That is, for every $n\le p_{N_0}$ and every $k\ge1$ such that $n+1,\ldots,n+k$ are all composite, there are distinct primes $P_1,\ldots,P_k$ with $P_i\mid n+i$ for $1\le i\le k$. The paper records (p. 1) that $p_{N_0}=19236701629>1.9\times10^{10}$.

Corollary 0.1 (p. 2). If $n+1,\ldots,n+k$ are all composite and $n\le p_{N_0}$, then

$$\omega\bigl((n+1)\cdots(n+k)\bigr)\ge k.\qquad(1)$$

The paper calls the assertion that (1) holds for all $n$ and $k$ with $n+1,\ldots,n+k$ all composite a weaker version of Grimm's conjecture (p. 2).

Proof pointer

P. 2: the paper says that for Theorem 1 it suffices to prove Theorem 2, the case $n=p_N$, $k=p_{N+1}-p_N-1$ for $1<N\le N_0$. The reduction is not spelled out; it runs as follows. A run of composites $n+1,\ldots,n+k$ with $n\le p_{N_0}$ lies inside $p_N+1,\ldots,p_{N+1}-1$, where $p_N$ is the largest prime not exceeding $n$, so $N\le N_0$, and $N>1$ because $n+1$ composite forces $n\ge3$; distinct primes for that maximal run restrict to the shorter one. Corollary 0.1 follows because the $k$ distinct primes $P_i$ all divide the product.

Read depth

Claims checked: the setting, Theorem 1, the value of $p_{N_0}$ and Corollary 0.1 were read clause by clause on the page images of the print. The computation behind Theorem 2 was not rerun. Nothing here is independently reviewed.

Dependencies

Theorem 2 of the same paper.

Source. S. Laishram and T. N. Shorey, Grimm's conjecture on consecutive integers, Int. J. Number Theory 2 (2006), no. 2, 207--211, doi:10.1142/S1793042106000498; the edition read is named on the source card.

Bears on

• Problem 375: the problem asks whether the distinct primes exist for all $n,k\ge1$; Theorem 1 gives them for every $n\le p_{N_0}=19236701629$ and every $k$, and says nothing for larger $n$. The problem's claim page for this paper records it as a partial result.