Exercise 1.3.2 (Largest number of consecutive square-free positive integers)

Question

What is the largest number of consecutive square-free positive integers? What is the largest number of consecutive cube-free positive integers, where $a$ is cube-free if it is divisible by the cube of no integer greater than $1$?

Richard Ganaye · Accepted Answer

\begin{proof} One among four consecutive numbers is always divisible by $4$, thus is not square-free. Since $5, 6, 7$ are three square-free consecutive integers,  the largest number of consecutive square-free positive integers is 3.

\bigskip
 One among eight consecutive numbers is always divisible by $8$, thus is not cube-free. Since $9, 10, 11, 12, 13 , 14 , 15$ are seven cube-free consecutive integers, the largest number of consecutive cube-free positive integers is $7$.

\end{proof}

Exercise 1.3.2 (Largest number of consecutive square-free positive integers)

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Exercise 1.3.2 (Largest number of consecutive square-free positive integers)

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