Prove that there is a prime number p such that n < p < n! for all n <2. n is an integer.

I'm not allowed to use Prime number theorem or this kind of stuff. Should be simple, but you must think of it!

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- Dec 3rd 2008, 07:28 AMvincisonfirePrime number
Prove that there is a prime number p such that n < p < n! for all n <2. n is an integer.

I'm not allowed to use Prime number theorem or this kind of stuff. Should be simple, but you must think of it! - Dec 3rd 2008, 07:58 AMPaulRS
Let be the greatest prime, less or equal than , then can only be divisible by primes greater than ( see here), thus,let be a prime number satisfying then (1) ( since otherwise we'd contradict our definition of )

From it follows that if ( show by induction that for all )

Thus, since we have (2)

(1) and (2) complete the proof for - Dec 3rd 2008, 06:25 PMJaneBennet
For we can take Assume that

Suppose to the contrary there is no prime with

Let be all the primes less than or equal to and consider

is greater than 1 and not divisible by any of the primes hence must be divisible by a prime

It follows that But say

Hence This can only hold if the LHS is 0, i.e. This is impossible for (contradiction). - Dec 3rd 2008, 06:40 PMThePerfectHacker