K-th Prime Proofs & Co-Prime Numbers

1. Sep 23, 2010

vmx200

I am having a hard time making head way on two problems related to the k-th prime and one about co-primes that I would really appreciate some help and/or direction!

Prove that:
(let pk be the k-th prime)

and

Regarding co-primes... is there any way to find a set of four numbers that are coprime, but cannot be subsequently grouped into sets of three that are?

Again, thank you for your time and generosity in helping me out!

Last edited: Sep 23, 2010
2. Sep 23, 2010

hamster143

In the first problem, is the number on the right prime or composite? If it's composite, what can you say about its factors?

3. Sep 23, 2010

vmx200

Oh, sorry!
Uhmn... pk is the k-th prime, so pk + 1 (the right most term) would be a composite, I believe?

4. Sep 23, 2010

hamster143

I'm asking about the whole thing: 1 + the product of all primes up to pk.

5. Sep 24, 2010

Gokul43201

Staff Emeritus
I believe you are misreading the expression.

To clarify: $p_{k+1} \leq (p_1~p_2 \cdots p_k)+1$