Three problems on k-th primes and coprime sets

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vmx200
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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)
Inequality p_{k+1} ≤ p_1 p_2 ⋯ p_k + 1, where p_k denotes the k-th prime.


and

Typeset problem statement: Prove that p_k < 2^(2^(k-1)). From this, prove that π(x) > log2 log2 x where π(x) is the number of primes ≤ x.

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!
 
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Discussion

In the first problem, is the number on the right prime or composite? If it's composite, what can you say about its factors?
 
hamster143 said:
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?
 
I'm asking about the whole thing: 1 + the product of all primes up to pk.
 
vmx200 said:
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: [itex]p_{k+1} \leq (p_1~p_2 \cdots p_k)+1[/itex]