Conjecture for prime pairs of difference two

Click For Summary

Discussion Overview

The discussion revolves around the Twin Prime Conjecture, specifically whether the number of prime pairs with a difference of two approaches infinity. Participants explore the implications of this conjecture and related concepts in number theory.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants inquire whether the conjecture implies infinitely many primes separated by the same even number or by a multiple of the same even number.
  • Others mention that the latter interpretation is true, referencing the implications of the Elliott-Halberstam conjecture and the Goldston-Pintz-Yıldırım result.
  • One participant notes that the condition g_n > \sqrt{\log p_n} for all n sufficiently large has not been disproven, suggesting a level of uncertainty regarding unconditional results.
  • There is a clarification that the number of pairs of primes with a difference of two is indeed related to the Twin Primes Conjecture.
  • Another participant challenges the phrasing of "approaches" infinity, asserting that the number of twin primes must either be infinite or a specific integer.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the conjecture and the implications of related mathematical results. No consensus is reached regarding the phrasing of the conjecture or the nature of the prime pairs.

Contextual Notes

There are limitations regarding the assumptions made about the nature of prime gaps and the conditions under which certain results hold. The discussion reflects ongoing uncertainty in the field.

Loren Booda
Messages
3,115
Reaction score
4
Can it be proven that the number of prime pairs with a difference of two (that is, primes separated by only one even number) approaches infinity?
 
Physics news on Phys.org
This is the of-yet-unproved Twin Prime Conjecture.
 
Loren Booda said:
primes separated by only one even number

Do you mean that infinitely many primes separated by the same even number? Or do you mean infinitely many primes separated by a multiple of the same even number? The latter is true.
 
Dragonfall said:
Do you mean that infinitely many primes separated by the same even number? Or do you mean infinitely many primes separated by a multiple of the same even number? The latter is true.

I know Elliott-Halberstam implies that (via Goldston-Pintz-Yıldırım), but is it known unconditionally? As far as I know, [itex]g_n>\sqrt{\log p_n}[/itex] for all n sufficiently large has not been disproven.

Oh wait, I just reread what you wrote. The latter is trivially true, since all prime gaps but the first are divisible by 2.
 
Dragonfall said:
Do you mean that infinitely many primes separated by the same even number? Or do you mean infinitely many primes separated by a multiple of the same even number? The latter is true.

The number of pairs of primes with a difference of two.
 
That would be the said "Twin Primes Conjecture".
 
But you wouldn't say "approaches" infinity in either case. Either the number of twin primes is infinite or it is a specific integer.
 

Similar threads

Graduate Proving Goldbach's conjecture by induction (or why it doesn't seem to work)
  • · Replies 10 ·
Replies
10
Views
3K
Graduate Proof of the twin primes conjecture
  • · Replies 3 ·
Replies
3
Views
3K
High School Are All Imaginable Integer Series Necessarily Infinite?
  • · Replies 17 ·
Replies
17
Views
2K
Graduate An approach to the Twin Prime Conjecture
  • · Replies 14 ·
Replies
14
Views
6K
Graduate Is It Reasonable to Assume a Prime Gap of at Least 10 Before Twin Primes?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Localising a non integral domain at a prime
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Monumental Proof Settles Geometric Langlands Conjecture
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Fundamental theorem of arithmetic from Jordan-Hölder theorem
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Questions regarding Kurepa's Conjecture
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Is the Goldbach Conjecture part of a larger pattern?
  • · Replies 1 ·
Replies
1
Views
2K