Riesel base 6 - team drive #4 - EIGHT OR BUST!

[Batalov]

521-530K is done and attached.

[gd_barnes]

Originally Posted by Mini-Geek

Perhaps it's not a simple 'yes or no', but based on some other factors that we don't know to look for yet.

To that end, has someone done that sort of comparison yet? To see how many primes there are per unsieved candidate vs how many we think we should expect?

We started that at NPLB but didn't continue it. I'd really like to do it once we get all k<=1001 at n=1M; which I'm shooting for mid to late 2011 on. There has to be a highly significant # of tests on each k to make any comparison statistically significant. Personally, that was a big reason why I started the project. You can't do such comparisons with big holes in testing ranges.

[Mini-Geek]

Originally Posted by gd_barnes

We started that at NPLB but didn't continue it. I'd really like to do it once we get all k<=1001 at n=1M; which I'm shooting for mid to late 2011 on. There has to be a highly significant # of tests on each k to make any comparison statistically significant. Personally, that was a big reason why I started the project. You can't do such comparisons with big holes in testing ranges.

I'm not sure just how many are necessary to be "statistically significant", but I've done a bit of work and attached the results (using screen caps from my TI-83 Plus ). From the obvious trends visible in these graphs, I'd draw the following likely conclusions:
low weight k's have widely varying weight/prime ratios, probably due to the granularity of the number of primes they get
higher weight k's have more consistent weight/prime ratios, probably due to the large number of primes present
there is no significant difference in how many primes are found per candidate in low vs high weight k's

(all data pulled from http://www.rieselprime.de/Data/00300.htm today, the two graphs include only k's that have that exact max n, not just at least that max n. This was done so I wouldn't have to mess with removing n values within a range from a k's sum of primes. The "weight/primes" is calculated from the Nash weight divided by the number of primes up to the search limit)
Note that the very low r values suggests a lousy linear correlation, as the graph confirms. However, overall from looking at it, I'd say an accurate best fit line is probably more like y=b than y=ax+b. With b about 57 plus or minus 3.


These findings might make sense of the fact that some low weight k's (or groups of them, as we always encounter in CRUS-like work, including FoB and SoB) seem to be unlucky, and others seem to be extremely lucky. It's just that the cone spreads out, and sometimes you're caught in the upper end of that, (low weight/very low primes) and sometimes in the lower end of it (low weight/many primes).
Just about the only other question is if there's any rhyme or reason as to which end of the cone a k will end up in...

[mdettweiler]

Originally Posted by Mini-Geek

Just about the only other question is if there's any rhyme or reason as to which end of the cone a k will end up in...

Right, indeed. Without any hard mathematical data to back me up, it does seem that certain bases, in particular, tend to be better in this regard than others. Riesel base 6 and Dual-Sierpinski base 2, to use the examples mentioned above, would fit this category. Others, for example, like Riesel base 23, have stuck on for what seems like forever on their final k, in this case much to my chagrin since I've taken it all the way up to 570K (pretty much the reasonable edge of my capabilities) from 226K with absolutely no primes.

The thing is, though, this seems to be a second trend not necessarily related to the primeness of the base, which usually affects conjecture size more than anything. Take the bases 3, for that matter. They are very prime, but as such have huge conjectures. Not so for Riesel base 6; Gary, I think you said base 6 is rather composite, right? Yet we're finding primes at an extremely "luckier" rate than we'd expect statistically. Sierp. base 6 is less particularly impressive; if anything, from gut observations of it I'd say we're in fact under the expected number of primes. So this presumed trend may differ with regard to Riesel/Sierp. side of a base, lending further credence to the idea that it's independent of the primeness of the base.

But, hey, I could be completely off here--just some random musings.

[gd_barnes]

Originally Posted by mdettweiler

Right, indeed. Without any hard mathematical data to back me up, it does seem that certain bases, in particular, tend to be better in this regard than others. Riesel base 6 and Dual-Sierpinski base 2, to use the examples mentioned above, would fit this category. Others, for example, like Riesel base 23, have stuck on for what seems like forever on their final k, in this case much to my chagrin since I've taken it all the way up to 570K (pretty much the reasonable edge of my capabilities) from 226K with absolutely no primes.

The thing is, though, this seems to be a second trend not necessarily related to the primeness of the base, which usually affects conjecture size more than anything. Take the bases 3, for that matter. They are very prime, but as such have huge conjectures. Not so for Riesel base 6; Gary, I think you said base 6 is rather composite, right? Yet we're finding primes at an extremely "luckier" rate than we'd expect statistically. Sierp. base 6 is less particularly impressive; if anything, from gut observations of it I'd say we're in fact under the expected number of primes. So this presumed trend may differ with regard to Riesel/Sierp. side of a base, lending further credence to the idea that it's independent of the primeness of the base.

But, hey, I could be completely off here--just some random musings.

No, I could never have said that about base 6. It is a quite prime base. Think about it. Take a look at other bases whose conjectures are ~84687. How many have even close to 4 k's remaining? Even if you go back to n=~66K, you have 14 k's remaining. the only bases that are close are b=2^q-1 bases. Riesel base 31 is the closest example. With a conjecture of k=134718, it only has 7 k's remaining at n=120K and had 15 k's remaining at n=25K. Virtually all other bases with a similar-sized conjecture have many 10's or 100's of k's remaining.

What you are describing here on things like Sierp base 6 and base 22 is virtually complete randomness. Nothing more and nothing less. Some bases will get lucky and find primes at a slightly better than normal rate and others slightly worse. There's no such thing as trends in randomness. I conjecture that extremely high weight k's or bases may have a slightly higher than normal # of primes per candidate searched then extremely low weight k's or bases. But that's only conjecture and certainly can't be proven by false "trends" on 2 bases.

When I use the term "a very prime base" it's only because the base's k's have few small factors and hence far less n-values are sieved out, which results in more k's having a prime at lower n-values. It's not because we can expect any higher percentage of primes per remaining sieved candidate. In other words, we will be finding more primes for the base, not because the base is getting lucky, but because there is more opportunity for a prime to occur.

The very prime 2^q-1 bases interest me for this reason but for just that same reason, most of their conjectures are extremely high. Riesel 31 with a "reasonable" conjecture is the exception and is one base that may interest me most in the future. Bases where b==(1 mod 30) are also quite prime, as a general rule, but not quite as much as 2^q-1 bases. B==(1 mod 30) bases have interested me somewhat more because most of their conjectures are not so high. It just so happens that bases 31 and 511 are the only bases being (possibly) searched by this project that are both b=2^q-1 and b==(1 mod 30). But since base 511 is such a high base with such a high conjecture, it makes base 31 one of the most interesting bases of all in that regard.

One final thing: For the # of k's remaining, Sierp base 6 is finding primes at a virtually identical clip as Riesel base 6. You're forgetting that the Riesel side went almost an n=300K range without a prime! The two sides of all individual bases should be virtually identical in weight and hence "primeness" or "compositeness".


Gary

[mdettweiler]

Ah, okay. I guess I'd just assumed that any very prime base would necessarily have a huge conjecture like base 3--but there are exceptions to every rule, and it seems Riesel base 6 (and as you also mentioned, Riesel base 31) are those exceptions.

I still think that eventually something will turn up that shows primes to be more than just random distributions--some of these otherwise-coincidences seem a little too big to be that. But, yeah, I guess without further evidence it's hard to speculate much beyond the assumption of a random distribution, which does seem to predict the odds of finding a prime pretty well in the long run.

[mdettweiler]

Reserving 530K-531K.

530K-531K complete, no primes; results attached.

Reserving 531K-532K.

531K-532K complete, no primes; results attached.

Reserving 532K-533K.

532K-533K complete, no primes; results attached.

[unconnected]

Reserving 533-543.

[gd_barnes]

Unconnected reported in an Email that n=533K-543K is complete. Nothing to report.

[kar_bon]

a note for Riesel Base 6 in the Riesel-conjecture-overview page would be better to see, it's done here; in the overview it seems available!
perhaps other bases, too?

[gd_barnes]

Originally Posted by kar_bon

a note for Riesel Base 6 in the Riesel-conjecture-overview page would be better to see, it's done here; in the overview it seems available!
perhaps other bases, too?

A link is at the top of the Riesel reservations page. There's a link to all of our team drives on the respective reservations pages. If there's a separate reservations page for the base, then it's at the top of that one, like is the case for base 16 on both sides.

You have to check the reservations pages to see if bases are available. That is where base 6 is clearly marked in yellow with the exception of k=1597 for R6, which is available since we've already tested it to n=1M and that is all that the team effort has sieved it to at this point.