Conjecture for prime pairs of difference two

In summary, the discussion is about the Twin Prime Conjecture which states that there are infinitely many pairs of primes with a difference of two. It is currently unproven, but it is known unconditionally that there are infinitely many primes separated by a multiple of the same even number. The number of pairs of twin primes is either infinite or a specific integer, so saying that it "approaches" infinity is not accurate.
  • #1
Loren Booda
3,125
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?
 
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  • #2
This is the of-yet-unproved Twin Prime Conjecture.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
That would be the said "Twin Primes Conjecture".
 
  • #7
But you wouldn't say "approaches" infinity in either case. Either the number of twin primes is infinite or it is a specific integer.
 

1. What is the "Conjecture for prime pairs of difference two"?

The "Conjecture for prime pairs of difference two" is a mathematical conjecture that states there are an infinite number of prime numbers that are exactly two units apart, such as 41 and 43, or 71 and 73.

2. Who proposed this conjecture?

The "Conjecture for prime pairs of difference two" was first proposed by French mathematician Alphonse de Polignac in 1849.

3. Has this conjecture been proven?

No, this conjecture has not yet been proven. It is still an open problem in mathematics.

4. Why is this conjecture important?

This conjecture is important because it has connections to other unsolved problems in mathematics, such as the twin prime conjecture, and its proof could potentially lead to a better understanding of prime numbers.

5. What progress has been made towards proving this conjecture?

Several mathematicians have made progress towards proving this conjecture, including Yves Gallot, who showed that there are infinitely many pairs of primes that differ by exactly two units and are also the same distance from the nearest perfect square.

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