Laishram 2006 grimm s conjecture consecutive integers

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theorem_1: Laishram and Shorey's verification that Grimm's conjecture holds for every n <= p_{N_0}, where N_0 = 8.5 x 10^8 and p_{N_0} = 19236701629, and for every k, with the consequence that omega((n+1)...(n+k)) >= k whenever n+1, ..., n+k are all composite and n <= p_{N_0}.

theorem_2: Laishram and Shorey's verification of Grimm's conjecture on each maximal run of composites between consecutive primes, n = p_N and k = p_{N+1} - p_N - 1, for 1 < N <= N_0 = 8.5 x 10^8, which suffices for their Theorem 1.

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Laishram, Shanta and Shorey, T. N., Grimm's conjecture on consecutive integers. Int. J. Number Theory 2 (2006), no. 2, 207--211, doi:10.1142/S1793042106000498. The copy read for this card is an author preprint with no journal header, which prints no copyright or license line on any of its five pages; the author's publications page that lists the paper (https://www.isid.ac.in/~shanta/publications.html, read 2026-10-02) carries only the site footer "Copyright © Dr. Shanta Laishram | Website Designed by Design Futuristic", which speaks for the website, and states no license or terms for the papers; the term is unstated.

The paper gives a numerical verification of Grimm's conjecture, which asks for distinct primes P_i dividing n+i for 1 <= i <= k whenever n+1,...,n+k are all composite. Theorem 1 states the conjecture holds for all n <= p_{N_0} with N_0 = 8.5 x 10^8, and p_{N_0} = 19236701629 > 1.9 x 10^10, for every k; Corollary 0.1 deduces omega((n+1)...(n+k)) >= k in that range. Theorem 1 is reduced to Theorem 2, which proves the conjecture at n = p_N with k = p_{N+1} - p_N - 1 for 1 < N <= N_0, since the maximal composite runs lie between consecutive primes; Lemma 0.2 checks that these gaps satisfy k(N) < (log p_N)^2 (a Cramer-type bound) for N <= N_0. The proof combines Philip Hall's theorem on systems of distinct representatives with a Sylvester-Erdos argument bounding n < k^t, plus about a week of Mathematica computation on an Intel Xeon. Context recorded includes what the paper calls the best known result, of Ramachandra, Shorey and Tijdeman: for an absolute constant c_2 > 0, n >= 3 and g = [c_2 (log n / log log n)^3], distinct primes P_i | n+i exist for 1 <= i <= g; the paper notes that c_2 is very small, so this is valid only for large n, and Erdos's observation, cited from Erdos and Selfridge, that Grimm's conjecture implies p_{i+1} - p_i <= c_1 p_i^{1/2 - alpha} for some alpha > 0 and an absolute constant c_1 (p. 1). This is the computational verification cited for Problem 375.

Page numbers on this card and its result pages are those of the author preprint read, pp. 1--5, which correspond to pp. 207--211 of the journal. Read status: claims checked for Theorem 1 (p. 1), Corollary 0.1, Lemma 0.2 and Theorem 2 (p. 2), read clause by clause on the page images; the proof of Theorem 2 (pp. 2--5) followed for structure; the computations were not rerun. Nothing here is independently reviewed.

Source: https://www.isid.ac.in/~shanta/publications.html.

Bears on. #375: Theorem 1 (p. 1) gives the problem's distinct primes for every run of composites n+1, ..., n+k with n <= p_{N_0} = 19236701629 and any k, by way of Theorem 2 (p. 2); it decides nothing for larger n.

Results.

• Theorem 1 (p. 1) and Corollary 0.1 (p. 2): Grimm's conjecture holds for all n <= p_{N_0} = 19236701629 > 1.9 x 10^10 and all k; hence omega((n+1)...(n+k)) >= k when n+1, ..., n+k are all composite and n <= p_{N_0}.
• Theorem 2 (p. 2) and Lemma 0.2 (p. 2): Grimm's conjecture is valid for n = p_N and k = k(N) = p_{N+1} - p_N - 1 for 1 < N <= N_0 = 8.5 x 10^8, which suffices for Theorem 1; and k(N) < (log p_N)^2 for N <= N_0.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.