Consecutive Odd Square-Free Positive Integers

Number Theory Level 3

There are $k$ consecutive odd positive integers, each of which is square-free.

What is the maximum possible value of $k$ ?

Details and Assumptions:

A positive integer $n$ is square-free if there is no positive integer $k>1$ such that $k^2 \Big|\, n$ .

The answer is 8.

1 solution

Paul Hindess
Dec 31, 2016

Any multiple of 9 is not square-free as it is divisible by 9 = 3 $^2$ .

Between any two odd multiples of 9 there are 8 odd numbers, so this will be the longest "run" if we can find such a run in which all the numbers are square-free.

The run 29, 31, 33, 35, 37, 39, 41, 43 is square-free, so the maximum value of $k$ is 8.

Consecutive Odd Square-Free Positive Integers

The answer is 8.

1 solution

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