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I suppose there are a series of such questions, depending on the length of the set, staring with pairs, and the power that is supposed to be missing from the divisors.

Here is a problem insisting on the opposite; that the numbers should all have a certain power as divisor.

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up vote 5 down vote accepted

The same question has been asked and answered on MathOverflow:

http://mathoverflow.net/questions/59741/are-there-infinitely-many-triples-of-consecutive-square-free-integers

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Thanks, oddly enough my few searches did not locate that page. Is there an elementary answer for pairs of consecutive numbers? –  Maesumi Mar 3 '13 at 21:07

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