[QUOTE=George M;475977]I came across this on a Wikipedia Article an am trying to find a proof, but nonetheless, if this equation has its own article, then there must be a way of demonstrating the truth of this equation.
[/QUOTE]
It is trivial. For example with \(n=3\) as \(p_1=2\) we have
\[ p_4-p_3+p_3-p_2+p_2-p_1+p_1=p_4. \]

[QUOTE=CRGreathouse;475975]I wrote that page.[/QUOTE]

Very nice!

Helfgott finally published his sieve: [url]https://arxiv.org/abs/1712.09130[/url]

I like Dudley's 1983 Formulas for Primes paper, which is free online from MAA.

For sieves, Quesada and Van Pelt (1996) and Quesada (1992).

For primality proving, I think the BLS75 paper ([url]http://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf[/url]) is an important collection of state of the art methods in 1975, many of which are still used for special forms.

For primality testing there are lots of Frobenius variants:

general polynomial (Grantham)
standard quadratic (lower case q), Grantham and others including Crandall & Pomerance
standard cubic (Buell and Kimball outlined from Grantham)

^ the above don't specify the parameters, like Fermat and M-R have an unspecified base that's important in practice. I believe all the ones below include parameter selection so are more like BPSW in that the user doesn't add extra parameters.

QFT (upper case q) from Grantham with extra steps
EQFT, SQFT, MQFT variations on Grantham's QFT
Khashin (2013)
Underwood (2012, 2014, 2016?)

[QUOTE=danaj;476065]Helfgott finally published his sieve[/QUOTE]

I saw that two days ago, very exciting. He has much more detail than he gave in his talk.

[QUOTE=danaj;476065]

Helfgott finally published his sieve: [URL]https://arxiv.org/abs/1712.09130[/URL]
[/QUOTE]



Thanks for the link, very interesting.
About processing the primes it is faster to save the primes in to RAM, but if you want to work on the goldbach conjecture you will need at least a range of 10^12 per package. You can only use the segmented sieve, e.g. processing about 8 GB of primes, load them into RAM and after your memory is full you can create the goldbach partitions and prime gap analysis. Than you can proceed the next primes.
This is also way better for your SSD (if you would run this on SSD´s), the maximum write and read function is limited. (e.g. 728 Terabyte for an Hyper X 3K, 240 GB)
Such tasks can be done using the BOINC wrapper technology.