K-th Prime Proofs & Co-Prime Numbers

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SUMMARY

The discussion centers on two mathematical problems involving the k-th prime number and co-prime numbers. The first problem requires proving that the expression \(p_{k+1} \leq (p_1 \cdots p_k) + 1\) indicates that \(p_k + 1\) is composite. The second problem explores the existence of a set of four co-prime numbers that cannot be grouped into sets of three co-prime numbers. Participants clarify the definitions and implications of prime and composite numbers in relation to these problems.

PREREQUISITES
  • Understanding of prime numbers and composite numbers
  • Familiarity with co-prime numbers and their properties
  • Basic knowledge of mathematical proofs and inequalities
  • Experience with number theory concepts
NEXT STEPS
  • Research the properties of prime numbers and their distributions
  • Study the concept of co-primality and its applications in number theory
  • Explore mathematical proofs related to the k-th prime number
  • Investigate the implications of the product of primes in relation to composite numbers
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Mathematicians, students of number theory, and anyone interested in advanced mathematical proofs and properties of prime and co-prime numbers.

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

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