Are There Exceptions to the Goldbach Conjecture for Even Numbers of the Form 2p?

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Discussion Overview

The discussion revolves around the Goldbach Conjecture, specifically examining whether there are exceptions for even numbers of the form 2p, where p is a prime. Participants explore the representation of these numbers as sums of two different primes, focusing on both theoretical implications and computational findings.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • Some participants note that 4=2+2 and 6=3+3 are examples of even numbers of the form 2p that cannot be represented as the sum of two different primes.
  • Others suggest that there is almost surely no other case of such exceptions, although they acknowledge that a proof remains elusive.
  • A participant mentions a computer program that shows no exceptions up to a limit of one hundred thousand.
  • Another participant claims to verify the absence of further exceptions up to approximately 9 × 1017, referencing a specific sequence for validation.
  • It is noted that according to Tomás Oliveira e Silva, the bound for exceptions is confirmed up to 1.609 × 1018.

Areas of Agreement / Disagreement

Participants generally agree on the absence of exceptions up to certain computational limits, but there is no consensus on the existence of exceptions beyond those limits, and the proof of the conjecture remains unresolved.

Contextual Notes

The discussion relies on computational findings and theoretical conjectures, with limitations regarding the proof of the conjecture and the bounds of the computational checks mentioned.

mathman
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4=2+2, 6=3+3. Are there any other cases where an even number of the form 2p, where p is a prime, cannot be represented as the sum of two different primes?
 
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mathman said:
4=2+2, 6=3+3. Are there any other cases where an even number of the form 2p, where p is a prime, cannot be represented as the sum of two different primes?

There is almost surely no other case, but a proof is far beyond reach.
 
A quick&dirty computer program shows no cases up to a hundred thousand.
 
I can verify that there are no further exceptions up to http://www.research.att.com/~njas/sequences/A025018 (67) = 906030579562279642 ≈ 9 × 1017.

Edit: According to Tomás Oliveira e Silva, this bound is good up to 1.609 × 1018.
 
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