Some Somewhat Easier n^i Sequences Available for Termination

[richs]

1352^51 terminated by Ed and me.

[gd_barnes]

As promised, below are updated lists of bases with the fewest and most same-parity sequences remaining. (All-parity sequences for double-square bases.) I included the recently initialized bases in this thread even though they are not on the pages yet under the assumption that they will be added in a manner similar to what was previously discussed.

For 1 remaining format is:
Base (size/cofac/smallest fac)

**********
Initialize size <= 180 digits:
**********
Fewest:

1 remain:
127 (173/152/113567)
167 (147/144/6469)
14536 (158/152/5)
9699690 (168/146/7)
200560490130 (160/126/3)

2 remain:
37, 42, 55, 59, 151, 239, 385, 552, 564, 660, 2310, 12496, 14316, 131071, 510510, 6469693230, 8589869056

Most:

11 remain:
200
10 remain:
28, 30
9 remain:
18, 78, 94, 882
8 remain:
40, 68, 88, 105

**********
All sizes:
**********
Fewest:

1 remain:
14536 (158/152/5)
9699690 (168/146/7)
33550336* (181/143/3)
200560490130 (160/126/3)
7420738134810* (182/161/3)
* - These bases are completed for <= 180 digits.

2 remain:
37, 42, 55, 59, 385, 552, 564, 660, 2310, 12496, 14316, 131071, 510510

Most:

14 remain:
94
12 remain:
78, 88, 92, 191, 200
11 remain:
72, 86, 87, 90, 91, 93, 95, 96, 99, 105, 199

[gd_barnes]

Originally Posted by EdH

<snip>
2) (I think) one item Jean-Luc is interested in, is sequences that terminate with the base prime. All of the primes less than 250 have tables so we're already working them. There are 11 sequences remaining that are in these tables and are at a term less than 150 digits. They are:

29^109
163^69
167^77
173^71
179^71
191^65
199^65
227^65
229^65
239^73
241^63

Some are already below 145. I'll focus on bringing them all below 135.

I think all the remaining primes would be too high to work their bases right now. The lowest (other than 181 that Jean-Luc has reserved) is 251.

I like this idea. Looking at the sequences remaining list that I just posted, 167^77 would complete that base for <= 180 digits. 239^73 would bring it down to only 1 sequence remaining for <= 180 digits. The prime bases are certainly the best performing.

One that you could add to the list is 14536^38. At 158/152 with a smallest factor of 5, it would be doable. It's not a prime base but is the easiest one remaining that would fully complete a base, regardless of size.

Why would prime bases > 250 be too high? Prime bases have been shown to have same-parity sequences that terminate more easily.

3) (I think) Jean-Luc is also interested in sequences that end in cycles, but studying the ones we already know, doesn't provide any indication of how to find more, so the only way is to work more sequences. We can work more sequences by working lower exponents, but we don't want to randomly poke around such that we can't easily create a fully initialized table.*

I agree that bases need to be fully initialized to be added to his page.

This brings us back to working at lower bases. All lower uninitialized bases are now composite, so this work won't satisfy #1 above, but we can get more matched parity sequences done and work toward turning up more cycles. You noted 102 in one of your posts, so let's initialize it (mixed parity to 110) and work on all matched parity sequences below 150 digits. <snip>

That would work. We could divvy up the work between you, Rich, and me however we see fit on both parities. As you you stated later on, we could then evaluate how it is working.

You mentioned about helping the project stay organized. A couple of things might help:
(1) On the web page, separate the opposite and same-parity sequences with regard to the counts and percentages that are complete by base. That would tell us more accurately what remains on a base and how it is performing vs. its neighbors.
(2) Have a standardization of the size that a base is tested to. This would give us more of an apples-to-apples comparison of everything that would make all of the count/percentage stats more accurate. It could even give us a heads up about anomalies in the performances of certain bases when compared to their neighbors that might lead Jean-Luc onto something in the data harvest.

[gd_barnes]

29^109 and 163^69 terminate

[EdH]

Originally Posted by gd_barnes

. . .
One that you could add to the list is 14536^38. At 158/152 with a smallest factor of 5, it would be doable. It's not a prime base but is the easiest one remaining that would fully complete a base, regardless of size.

Why would prime bases > 250 be too high? Prime bases have been shown to have same-parity sequences that terminate more easily.
. . .

I'll add 14536^38. That one might take a bit to fall. I was thinking >250 would present larger sequences quickly and produce much fewer new terminations within a given time frame, but if matched base terminations are preferred, that's the only place we'll find more. OTOH, the lower bases will give us more chances to find cycles, but only finding one or two more cycles may not add any insight for others.

Originally Posted by gd_barnes

(1) On the web page, separate the opposite and same-parity sequences with regard to the counts and percentages that are complete by base. That would tell us more accurately what remains on a base and how it is performing vs. its neighbors.
(2) Have a standardization of the size that a base is tested to. This would give us more of an apples-to-apples comparison of everything that would make all of the count/percentage stats more accurate. It could even give us a heads up about anomalies in the performances of certain bases when compared to their neighbors that might lead Jean-Luc onto something in the data harvest.

All web page work would fall on Jean-Luc and Karsten, so that would need to be brought up to them. I was kind of referring to our current work not getting too difficult for Jean-Luc to incorporate when he returns.

[EdH]

I think 180 is a bit too high for the current cutofff. I think 150 might be better for now. That way they can be brought down relatively quickly. I'm even reconsidering the one I just said I'd add (14536^38). That could easily turn into a few days to complete. I won't be doing much to help with initialization if I'm running other sequences. Let's revisit base completion after we do some other work.

Are we looking at 102 or 251 as a base to initialize?

[EdH]

It looks like I'll be tied up for a bit. The list I mentioned will still be underway and I'm still thinking about 14536, but work as you would prefer and I'll try to catch up later.

[gd_barnes]

Originally Posted by EdH

I think 180 is a bit too high for the current cutoff. I think 150 might be better for now. That way they can be brought down relatively quickly. I'm even reconsidering the one I just said I'd add (14536^38). That could easily turn into a few days to complete. I won't be doing much to help with initialization if I'm running other sequences. Let's revisit base completion after we do some other work.

Are we looking at 102 or 251 as a base to initialize?

On for a little while. Back on longer in a few hours.

I was under the impression that we wanted to come as close as we can to fully completing bases; hence the 180-digit cutoff. That said, I like your idea a whole lot better! Slowly working our way up a size list by base in this way seems a lot more doable. We could even go 150 until all complete, then 155, then 160, etc. It would give us plenty of interim base completions along the way that are not so difficult.

At this point, if I made a list of sequences remaining by base that have a beginning start point of <= 150 digits, the list would likely have very little on it. I have lists of counts of bases/sequences separated with a start point of both <= 160 digits and 180 digits. I'll see what I can conger up for <= 150 digits. If it's almost nothing left, I can fairly quickly prepare one for <= 160 digits that could prove more useful.

What about initializing base 120? I've seen that one mentioned a few times around here. I think I'd prefer that one instead of base 102 or 251. With lots of small factors in the base, it might have some interesting properties.

[gd_barnes]

137^79, 167^77, and 392^62 terminate

[EdH]

Base 120 was brought up by sweety a few posts back in the other thread. That exchange was the only mention I recall. It's certainly doable. RichD is working on 104 because I didn't know the status of 102. I'd like to fill that gap in also. Since 120 is of interest, let's initialize it to see what happens. Then, maybe we can fill in 102. Let's see what your next assessment shows. We may want to initialize the mixed parity before terminating the matched for some of these.

[gd_barnes]

I added a column to my spreadsheet for starting size and got it populated for all sequences starting at <= 160 digits. That helps for what we are looking for. Below is what I came up with.

To the best of my knowledge, this is up to date as of this post.

We have completed all sequences with a starting size at < 143 digits.

Remaining for starting size <= 145 digits (last number is smallest factor):

20^110: 144/140/3
78^76: 144/138/5
94^72: 142/142/3
306^58: 145/140/59

Remaining for starting size 146-150 digits:

20^112: 144/134/3
20^114: 148/129/3
21^113: 150/147*/3
24^108: 150/144*/5
28^102: 148/142*/3
87^77: 144/141/3
94^76: 143/128/3
191^65: 144/139/3
199^65: 144/136/5
306^60: 148/139/7
1184^48: 148/145*/3
1210^48: 148/141*/3
14264^36: 144/131/3
14288^36: 148/141/7

* - ECM'd to t40 (All others fully ECM'd.)

Breaking it down in a count format by base like before for all <= 150 digits:
1 remaining:
21, 24, 28, 78, 87, 191, 199, 1184, 1210, 14264, 14288
2 remaining:
94, 306
3 remaining:
20