# Solution: 2008-1 Distinct primes

Chiheon Kim (김치헌)

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!

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 $$n<19000000000$$.