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Prove that, for every k > 1, there is a n such that each of n+1,...

Prove that, for every k > 1, there is a n such that each of n+1, n+2, ···, n + k is not a prime number.

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Answer #1

We can directly give an expression for such an n (with justification, of course)

We can say that works because which is not prime as it is divisible by

And in general, we have

For

Each of which is a divisor of

Hence, is divisible by the divisor and so it cannot be a prime

Thus, we can choose such that are all not primes

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