Let \(n\) be a positive integer. Let \(a_1,a_2,\ldots,a_k\) be distinct integers larger than \(n^{n-1}\) such that \(|a_i-a_j|<n\) for all \(i,j\).Prove that the number of primes dividing \(a_1a_2\cdots a_k\) is at least \(k\).

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Click here for his solution of Problem 2008-1.

This problem is equivalent to a theorem of Grimm (see his paper, A Conjecture on Consecutive Composite Numbers, *The American Mathematical Monthly*, Vol. 76, No. 10 (Dec., 1969), pp. 1126-1128). He conjectured that the same thing can be done without the lower bound \(n^{n-1}\). Laishram and Shorey verified Grimm’s conjecture when \(n<19000000000\).

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