K-th Prime Proofs & Co-Prime Numbers

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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)
Picture1-1.png


and

Picture3-2.png




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

Gokul43201

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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]
 

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