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Find the largest positive integer $k$, such that $\mu(n+r)=0$ for all $1\leq r\leq k$ where $r,n$ are positive integers.

As far as I could make out, we need to find out the maximum range(if nay) of numbers where each has a square divisor.

I have gone through the theory of square-free numbers here and there, but could not proceed much.

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    Depe$n$ds on $n$ in a chaotic way, usually smallish, but can be arbitrarily large.2012-07-24

1 Answers 1

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There is no such largest positive integer. Given $k$ primes, by the Chinese remainder theorem we can find a number $m$ that has remainders $1$ through $k$ with respect to their squares. Then $m-k$ through $m-1$ all have square divisors.

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    Agreed & accepted.2012-07-25