Timeline for Yitang Zhang's paper
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 15, 2014 at 19:59 | comment | added | GH from MO | @John Nicholson: The first 54 primes do not form an admissible tuple. To get a convenient choice of an admissible $k$-tuple, take the first $k$ primes beyond $k$. | |
| Apr 10, 2014 at 22:36 | comment | added | John Nicholson | One more thing with $p_i$, the is any integer value $i > k$. | |
| Apr 10, 2014 at 22:06 | comment | added | John Nicholson | $p_k = R_n$ is the nth Ramanujan prime. | |
| Apr 10, 2014 at 22:01 | comment | added | John Nicholson | 251 is the 54th prime, so this 51-tuple will have most of the primes included. This seems similar to A165959 at the OEIS. This sequence does not fix k, but fixes the number of primes in the interval $[p_{i-n}, p_k]$ to n and allows one to find the next prime $p_i < 2*p_{i-n}$. What I am really wondering is if there is a way to combine these two ideas? It would be really cool if someone could prove that there are an infinite number of 3 in the sequence because this would prove the twin prime conjecture. The prime $p_i$ is then next prime after $R_n$ and $p_{i-n}$ is the next one after $R_n/2$. | |
| Apr 10, 2014 at 21:58 | comment | added | Stijn | That makes sense. If we insist on a lower bound on the distance between the elements of the admissible sets, say that $h_{n+1} - h_n \geq k$, are there any known general results for the diameter of the minimal admissible sets, that you're aware of? Something asymptotic, maybe? | |
| Apr 10, 2014 at 20:18 | comment | added | GH from MO | @Stijn: Actually there are weaker but effective versions of Pintz's result. So probably we know that there is a number between 252 and 70 million that occurs infinitely often as a difference of two primes. For that, all we need is an admissible 51-tuple whose pairwise distances fall between 252 and 70 million. | |
| Apr 10, 2014 at 17:28 | comment | added | Stijn | Oops, I referenced the wrong Pintz paper, it should have been "Polignac numbers, conjectures of Erdős on gaps between primes, arithmetic progressions in primes, and the bounded gap conjecture" which is on the arXiv, but that says that there is an ineffective constant $C$ such that the interval $[M, M+C]$ always contains some number k with the property that primes will infinitely often differ by precisely k. So we don't know if there is some Polignac number between 252 and 70 million but there is certainly a Polignac number above 252. | |
| Apr 10, 2014 at 16:30 | comment | added | John Nicholson | OK, so Zhang proof does not touch or cover the values greater than 70 million, but the Pintz paper does. I guess with the current reduction to 252 it is the Pintz paper doing the work for 252 to 70 million. | |
| Apr 10, 2014 at 15:03 | history | answered | Stijn | CC BY-SA 3.0 |