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2020-07-15, 22:11   #23
mart_r

Dec 2008
you know...around...

60810 Posts

No major news in the last few weeks, but in the attachment there's the data for CSGmax up to p=2n for n=[16..41] and q<=2690. Data for p=242 might follow next week. Minor differences in conduct compared to the data for the older analysis on 2020-05-01: here the endpoints p' of the gaps are key for clustering in powers of two, and CSG is determined using the Ri-sum-of-derivatives formula as first considered in post # 2 - has anyone got a catchy name for it, perchance?

$Ri(x)=1+\sum_{n=1}^\infty \frac{\log^nx}{n\hspace{1}n!\hspace{1}\zeta(n+1)}+\frac{\arctan\frac{\pi}{\log x}}{\pi}-\frac{1}{\log x}$

$CSG=\frac{q}{k\hspace{1}\varphi(q)}[\sum_{i=1}^k Ri'(p+iq)]^2$
(where k=(p'-p)/q - in other words, there are k-1 consecutive composites in the arithmetic progression p+i*q)

I get some more or less meaningful numbers out of the data, but nothing that would justify a new conjecture or other breakthrough. Or is it just me being cautious? Maybe someone else has a better idea how to commercialize the data.

Right now I'm unable to decide whether or not to believe that CSG has a global maximum, I mean, as (p,q) $\rightarrow \infty$. (Of course maths is not about beliefs, but a good deal of conjectures were or are based merely on beliefs, also known as SWAGs...)

With already more than 500 extraordinarily large gaps, there's still no second example where q and r are both prime, but I'm close, and confident that I'll find at least one more of those before the end of July. There should be on average one in about 150 in the range I'm searching.

There are two values of q for which two extraordinarily large gaps are known:
q=28388, r=5859 and r=11949 (p=5088100651 and p=366870073)
q=389104, r=88289 and r=258931 (p=461954737 and p=2176128499)
Also, and quite unusually, there's one value of r with two values of q:
r=197077, q=223808 and q=502458 (p=2317057493 and p=267504733)
Attached Files
 PGAP_data_CSG.zip (611.8 KB, 57 views)