Proving Sum of Two Primes is Never Twice a Prime

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Discussion Overview

The discussion revolves around the question of whether the sum of two consecutive prime numbers can ever equal twice a prime number. Participants explore definitions, provide examples, and reference previous discussions on related topics.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • One participant initially poses the question about the sum of two primes being twice a prime, but later clarifies that the intended question is about consecutive primes.
  • Another participant questions the definition of "twice a prime."
  • A counterexample is provided where the sum of two instances of the prime number 2 equals 4, which is twice 2, suggesting a potential flaw in the original question.
  • Further, a participant notes that the question may be redundant, referencing an earlier discussion about the average of two consecutive odd primes not being prime.
  • A mathematical argument is presented that assumes two consecutive primes and concludes that their average cannot be prime.

Areas of Agreement / Disagreement

Participants express differing views on the clarity and relevance of the question, with some suggesting redundancy and others providing mathematical reasoning. No consensus is reached regarding the validity of the claims made.

Contextual Notes

There are references to earlier discussions and assumptions about the nature of prime numbers and their sums, but these are not fully resolved within the current thread.

Who May Find This Useful

This discussion may be of interest to those exploring properties of prime numbers, mathematical proofs, or the relationships between consecutive primes.

sachinism
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Prove that sum of two primes can never be twice a prime

p.s: find the actual edited q in 4th post belowsorry for the mistake
 
Last edited:
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'twice a prime' ?
 
sachinism said:
Prove that sum of two primes can never be twice a prime

Counter examples: 2 (a prime) + 2 (a prime) = 4 = twice 2 (a prime); or if you object to using the same prime twice or more: 7 + 19 = 26 = 2 x 13
 
Last edited:
ah my bad

this is the correct q

Show that sum of two consecutive primes is never twice a prime
 
What is the difference between consecutive primes?
 
It seems to me that this question is redundant of an earlier thread on November 12which depended upon the fact that the average of two consecutive odd primes can not be a prime.
 
Last edited:
@ramsey

can you give me the link of that thread please
 
Let (j, k)\in\mathb{N}^2. Without loss of generality, assume j < k.


From here, it's safe to assume that: \forall (j, k), j < \frac{j + k}{2} < k.


Then, p_n < \frac{p_n + p_{n+1}}{2} < p_{n+1}.


Since p_n and p_{n+1} are consecutive primes, \frac{p_n + p_{n+1}}{2} cannot be prime.
 

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