K-th Prime Proofs & Co-Prime Numbers

[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!

 

[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?

 

[vmx200]

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

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

 

[hamster143]

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

 

[Gokul43201]

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

I believe you are misreading the expression.

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