Basis of estimate
 

Here is a screen capture of the LibreOffice formulas and input data used to compute the Oct 7 estimate.

[B]From [URL]https://www.mersenne.org/report_milestones/[/URL]
[/B]

[B]Progress toward next GIMPS milestones [I](last updated 2021-08-11 16:15:11 UTC, updates every 15 minutes)[/I][/B]

[LIST][*] All exponents below [COLOR=blue]56 606 299[/COLOR] have been tested and verified.[*] All exponents below [COLOR=blue]104 125 811[/COLOR] have been tested at least once.[*] Countdown to first time checking all exponents below 105 million: [URL="https://www.mersenne.org/assignments/?exp_lo=104125811&exp_hi=105000000&execm=1&exp1=1&extf=1&exdchk=1&excert=1"]94[/URL][*] Countdown to first time checking all exponents below 106 million: [URL="https://www.mersenne.org/assignments/?exp_lo=104125811&exp_hi=106000000&execm=1&exp1=1&extf=1&exdchk=1&excert=1"]6 689[/URL] [COLOR=black](4 722 available)[/COLOR][*] Countdown to verifying all tests below 57 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56606299&exp_hi=57000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]25[/URL][*] Countdown to verifying all tests below M(57 885 161): [URL="https://www.mersenne.org/assignments/?exp_lo=56606299&exp_hi=57885161&execm=1&exp1=1&extf=1&exfirst=1"]196[/URL][*] Countdown to verifying all tests below 58 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56606299&exp_hi=58000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]261[/URL][*] Countdown to verifying all tests below 59 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56606299&exp_hi=59000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]5 397[/URL] [COLOR=black](3 216 available)[/COLOR][*] Countdown to verifying all tests below M(74 207 281): [COLOR=blue]280 826[/COLOR][*] Countdown to verifying all tests below M(77 232 917): [COLOR=blue]336 300[/COLOR][*] Countdown to verifying all tests below M(82 589 933): [COLOR=blue]435 008[/COLOR][/LIST]Updated projection to determination of MP#48* as Mp#48 or not by exhaustive DC to that exponent level is ~2021-10-05, slightly earlier than the projection posted in late June.

[QUOTE=kriesel;585415]
Updated projection to determination of MP#48* as Mp#48 or not by exhaustive DC to that exponent level is ~2021-10-05, slightly earlier than the projection posted in late June.[/QUOTE]

My prediction is November 22, 2021
95%, ±82 days
50%, ±25 days

Let's see :)

An alternate way of estimating DC completion to Mp#48*
 

This is practical when the number of remaining exponents below a milestone is < ~200.
Go to [URL]https://www.mersenne.org/assignments/?exp_lo=56917213&exp_hi=57885161&execm=1&exp1=1&extf=1&exfirst=1[/URL]
Then sort by estimated completion date.
Then from the end of the list, latest estimated completion, consider that those with expiration date sooner than ETA will be reassigned if not completed before expiration.
Find the latest such expiration by moving back up the list from the end.
At the moment that's 57864227, 25 days from now.
There are currently 107 left below Mp#48*.
So that will be a Cat 0 DC upon expiration.
Cat 0 typically will complete within 7 days of assignment. (I expect reassignment to be swift, therefore negligible.)
So this initial approach predicts completion to Mp#48* by 25 + 7 = 32 days from 2021-08-21; by 2021-09-21.
Next, review the list for others that expire later than that, not currently updating. There are a few now up to 34 days and showing no current updating.
These will add up to 34 - 25 = 9 more days to the estimate, or ~ 2021-09-30, assuming no further updating or completion occurs before expiration.
There are a few with expiration dates further out, but these are currently updating.
Except for [URL]https://www.mersenne.org/M57817769[/URL] which last updated 2021-08-19 and has 35 days remaining. Corresponding to reaching Mp#48* by ~ 2021-10-01.
That is very close to the 2021-10-07 date estimated in June, 2021-10-05 estimated 11 August, or 2021-10-04 estimated 21 August, that were based on the 56M milestone trajectory and remaining # of exponents to Mp#48* versus date/time.

I believe the method of relying for projection fitting on the lowest unverified exponent over time is using an inferior metric, both because in a sense it depends at any point in time upon only a single exponent's state, and because those are consistently exceptional, laggard exponents.
It does not reflect the progress made upon the many other exponents that also must be cleared to reach a milestone.
In the extreme case, the lowest unverified exponent could remain fixed for its entire expiration period, up to ~60 days before completion or reassignment, if it is assigned as a cat 1 DC as part of a group of >200 remaining, to a system that begins it and then ceases transmitting updates or making sufficient progress, until it is the last remaining below the milestone.
Unlikely, but possible.

From [URL]https://www.mersenne.org/report_milestones/[/URL]

[B]Progress toward next GIMPS milestones [I](last updated 2021-08-21 14:15:08 UTC, updates every 15 minutes)[/I][/B]

[LIST][*] All exponents below [COLOR=blue]56 917 213[/COLOR] have been tested and verified.[*] All exponents below [COLOR=blue]104 244 157[/COLOR] have been tested at least once.[*] Countdown to first time checking all exponents below 105 million: [URL="https://www.mersenne.org/assignments/?exp_lo=104244157&exp_hi=105000000&execm=1&exp1=1&extf=1&exdchk=1&excert=1"]35[/URL][*] Countdown to first time checking all exponents below 106 million: [URL="https://www.mersenne.org/assignments/?exp_lo=104244157&exp_hi=106000000&execm=1&exp1=1&extf=1&exdchk=1&excert=1"]4 390[/URL] [COLOR=black](2 510 available)[/COLOR][*] Countdown to verifying all tests below 57 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56917213&exp_hi=57000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]7[/URL][*] Countdown to verifying all tests below M(57 885 161): [URL="https://www.mersenne.org/assignments/?exp_lo=56917213&exp_hi=57885161&execm=1&exp1=1&extf=1&exfirst=1"]107[/URL][*] Countdown to verifying all tests below 58 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56917213&exp_hi=58000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]138[/URL][*] Countdown to verifying all tests below 59 million: [URL="https://www.mersenne.org/assignments/?exp_lo=56917213&exp_hi=59000000&execm=1&exp1=1&extf=1&exfirst=1&excert=1"]4 326[/URL] [COLOR=black](2 752 available)[/COLOR][*] Countdown to verifying all tests below M(74 207 281): [COLOR=blue]278 624[/COLOR][*] Countdown to verifying all tests below M(77 232 917): [COLOR=blue]334 160[/COLOR][*] Countdown to verifying all tests below M(82 589 933): [COLOR=blue]432 867[/COLOR][/LIST]

I took a whack at estimating when the exponents up to 57885161 would all be double-checked in [url=https://www.mersenneforum.org/showpost.php?p=512300&postcount=3064]this March 31, 2019 post[/url]. My guess, based on simple extrapolation of then-current progress, was about two and a half years from then. The responses (thanks to all!) [and which you can see by clicking on [u] Newer milestone thread[/u] at top right of the single post view] indicated that my guess was not unreasonable.

One thing that has changed since then is that first-time LL testing is no longer routine. Instead, PRP testing which can be verified relatively cheaply is the order of the day.

[QUOTE=Dr Sardonicus;586254]I took a whack at estimating when the exponents up to 57885161 would all be double-checked in [URL="https://www.mersenneforum.org/showpost.php?p=512300&postcount=3064"]this March 31, 2019 post[/URL]. My guess, based on simple extrapolation of then-current progress, was about two and a half years from then. The responses (thanks to all!) [and which you can see by clicking on [U] Newer milestone thread[/U] at top right of the single post view] indicated that my guess was not unreasonable.

One thing that has changed since then is that first-time LL testing is no longer routine. Instead, PRP testing which can be verified relatively cheaply is the order of the day.[/QUOTE]Well done. 2.5 years from 2019-03-31 is 2021-09-30. Very good agreement with most recent estimates. 2021-11-22 is the outlier.
(This reminds me of a combustion model discussed decades ago in one of my advanced engineering classes. It incorporated so many independent simplifying assumptions, over two dozen. that their errors canceled well and it was quite accurate in most cases.)

PRP first testing began at p~75.5M so was not a factor in reaching 57885161 in first tests. PRP/GEC/proof can be used in place of LL DC and essentially eliminates the possibility of need for TC then. That favors earlier completion dates for DC to 57885161. TC would probably add only a few days to completion date.

From [URL]https://www.mersenneforum.org/showpost.php?p=512475&postcount=3072[/URL] posted 2 April 2019:
"I expect that ll tests below 47 mio will be verified on 21 April 2019 ±45 days
48 mio -- on 19 July 2019 ±45 days
49 mio -- on 14 October 2019 ±45 days
50 mio -- on 07 January 2020 ±45 days."
Actual milestone dates:
47 2019-04-07 -14 days
48 2019-09-05 +49 days
49 2019-11-19 +36 days
50 2020-02-23 +47 days
±45 days (~95% confidence) indeed. Although it didn't make much sense applying -45 days to 21 April 2019 in a 2 April 2019 post.

[QUOTE=kriesel;586257]
Actual milestone dates:
47 2019-04-07 -14 days
48 2019-09-05 +49 days
49 2019-11-19 +36 days
50 2020-02-23 +47 days
±45 days (~95% confidence) indeed. Although it didn't make much sense applying -45 days to 21 April 2019 in a 2 April 2019 post.[/QUOTE]

You're right. But all this was made by eye, with no particularly careful calculations.
And, it seems, this is even my very first post, which I made, when I still did not fully understand where I ended up :)

All exponent below 57M have been verified.
I predict that we'll finish the verification of M57885161 in about a month, probably before 2021-10-01.

I will correct my last prediction regarding All tests below M57885161 by 6 days,
from (November 22, 2021) to (November 16, 2021) ±25 days (50% probability tube).

[ATTACH]25548[/ATTACH]

Acceleration of All tests verification has been observed since August 19. If it stays at the current level, then there are chances that the kriesel's prediction regarding October 6 will turn out to be correct.

History extrapolation from 56M method: currently indicating ~2021-09-25 (which inherently includes allowance for number of exponents/day completing declining over time)

Alternate method update:
part one, of those exponents currently updating server, and not indicating expiration before estimated completion, latest estimated completion is 2021-09-20;
part two, laggards not currently updating, likely to expire, latest expiration date 18 days = Sep 26, + ~7 days for reassign & complete = 2021-10-03.
Completion is expected by about the later of the two parts' date estimates, 2021-10-03.

Chun sung soo has 9 not updating out of 32 remaining.

What we have at the moment.

[ATTACH]25638[/ATTACH]

I'm still sticking to my early prediction – November 16, 2021 ±25 days.
Maybe I will slightly correct it, but later, when I will more reliably understand the microtrend within the macrotrend.