Johnny Cheung wrote:

IF N=1, 2n,3N IS just adjanct, where can find one “between them”…

We should focus on the cases when n is large enough. It applies to most mathematical problems.

Let me restart the statement as ‘There is a prime p such that ‘.

Actually, there is an integer N such that the statement is correct for all n > N. This results simply come from Prime Number Theorem by Gauss:

Let denote the number of primes less than n. Then we have . ( is natural logarithm, i.e. logarithm with base e.)

I can’t tell what N is, but such N exists.

Besides, the number 1.5 in the statement can be replaced by any number k > 1. Of course, N needs to be very large if k is very close to 1. Anyway, such N exists.

When k = 2, it is the celebrated Bertrand’s postulate (it has been proved). I think your problem should be inspired from this theorem, right?

I don’t think the theorem can be proved easily with elementary number theory as the distribution of primes is not well discussed in elementary number theory._________________

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