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Show that the gap between two consecutive squarefree numbers can be arbitrary large.

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Problem 5. Recall that a number n is called squarefree if it is not divisible by any square > 1. Show that the gap between two consecutive squarefree numbers can be arbitrary large. (Hint: Find a positive integer m such that m is divisible by 2^2, m + 1 is divisible by 3^2, m + 2 is divisible by 5^2, m + 3 is divisible by 7^2 and so forth.)

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It is shown that the gap between two consecutive squarefree numbers can be arbitrary large. The solution is detailed and well presented.

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Proof:
Let , , , , and so on. Generally, is the th prime number. Now for any big integer , we consider the following system of ...

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