Predict M52

[Dr Sardonicus]

Not one, but two correct answers to my exercise!

Well, at least a little bit of good mathematical thinking came of the latest shambles of a prediction for M52...

[tuckerkao]

Originally Posted by Uncwilly

This is the list of currently active guesses (those stricken through were previous guesses that have been replaced under the rules).

Just checked the quoted post above, I'll use up the final guess on M198626023 with the date of June 23rd, 2022. "Final Answer" like in the "Who Wants to be the Millionaire Show", so UncWilly doesn't have to ask me again.

Originally Posted by slandrum

In an even base, if the last digit is even, then the number is even.
In an odd base, if the sum of the digits is even, then the number is even.

In any base b, the sum of the digits of the number is congruent to the number mod (b-1)

In a trine base, if the last digit is trine(divisible by 3), then the number is trine.
In a (trine + 1) base, if the sum of the digits is trine, then the number is trine.

What's the rule for (trine + 2) base? (3, 6, 11₈, 14₈, 17₈, 22₈, 25₈, 30₈)

[slandrum]

Originally Posted by tuckerkao

Just checked the quoted post above, I'll use up the final guess on M198626023 with the date of June 23rd, 2022. "Final Answer" like in the "Who Wants to be the Millionaire Show", so UncWilly doesn't have to ask me again.


In a trine base, if the last digit is trine(divisible by 3), then the number is trine.
In a (trine + 1) base, if the sum of the digits is trine, then the number is trine.

What's the rule for (trine + 2) base? (3, 6, 11₈, 14₈, 17₈, 22₈, 25₈, 30₈)

There's no one rule, there's a lot of ways to do it.

A simple way is to break the number into two digit pairs, if there's an odd number of digits, the first digit is by itself. Sum those numbers up, and the result will be the same mod 3 as the original number. You can repeat the process until you have a 2 digit number, then add twice the first digit to the 2nd digit, and that will also be the same mod 3 as the original number. Basically you are looking at the number in the base b², b² will be congruent to 1 mod 3.

[tuckerkao]

Originally Posted by slandrum

There's no one rule, there's a lot of ways to do it.

A simple way is to break the number into two digit pairs, if there's an odd number of digits, the first digit is by itself. Sum those numbers up, and the result will be the same mod 3 as the original number. You can repeat the process until you have a 2 digit number, then add twice the first digit to the 2nd digit, and that will also be the same mod 3 as the original number. Basically you are looking at the number in the base b², b² will be congruent to 1 mod 3.

I think it's called the "Alternating sums of the numerical blocks of size 1", must go from right to left if odd digits total, given 11₈ or 11 of any other (trine + 2) base have the trine character.

Dozenal divisibility of 5: Alternating sums of the numerical blocks of size 2. [Dozenal]25 * 5 = 101
Dozenal divisibility of 7: Alternating sums of the numerical blocks of size 3. [Dozenal]187 * 7 = 1001

[Dr Sardonicus]

Originally Posted by tuckerkao

<snip>
In a trine base, if the last digit is trine(divisible by 3), then the number is trine.
In a (trine + 1) base, if the sum of the digits is trine, then the number is trine.

What's the rule for (trine + 2) base?
<snip>

Nil sapientiae odiosius obscuritate nimia.

There should be a Forum rule against deliberately using gratuitously obscure lingo like "trine" when standard terminology (e.g. divisible by 3, congruent to 0 (mod 3)) exists.

Note that b^k == (-1)^k mod (b+1). Let d_k be the b^k base-b digit of n. Then

n == d₀ - d₁ + ... (mod b+1),

the alternating sum of base-b digits starting with the units digit.

For the decimal base 10 = ten, the alternating digit sum determines whether n is divisible by 11.

If b == 2 (mod 3) then 3|(b+1), so this alternating digit sum is congruent to n (mod 3).

[kriesel]

Originally Posted by Dr Sardonicus

There should be a Forum rule against deliberately using gratuitously obscure lingo like "trine" when standard terminology (e.g. divisible by 3, congruent to 0 (mod 3)) exists.

Added guidance to https://www.mersenneforum.org/showpo...64&postcount=2, linking back to the quoted post.

[tuckerkao]

Originally Posted by Uncwilly

We have a loooong way to go to say it will be the longest gap.

2013-01-25 - 2009-04-12 = 1384 days
1996-09-03 - 1994-01-04 = 973 <- sort of counts.
2015-09-17 - 2013-01-25 = 965
2001-11-14 - 1999-06-01 = 897 days

As of now we are 882 days

How about now?

Also, why M42643801 had the discovery date of Jun 4, 2009 not Apr 12, 2009?

[ATH]

Originally Posted by tuckerkao

How about now?

Also, why M42643801 had the discovery date of Jun 4, 2009 not Apr 12, 2009?

It was turned in to the primenet server on Apr 12th, which originally was considered the discovery date. But it was not noticed until June 4th which we later decided should be the real discovery date.

[ATH]

Here is the list using the agreed upon discovery dates, and today July 30th we reached the previous record gap of 1331 days, so tomorrow will be a new record This is not a record we want to beat.

Here is a "countup" timer since the discovery of M(82589933):
https://www.timeanddate.com/countdow...3&font=cursive

M(1398269)	Nov 13 1996
M(2976221)	Aug 24 1997	+284 days
M(3021377)	Jan 27 1998	+156 days
M(6972593)	Jun  1 1999	+490 days
M(13466917)	Nov 14 2001	+897 days
M(20996011)	Nov 17 2003	+733 days
M(24036583)	May 15 2004	+180 days
M(25964951)	Feb 18 2005	+279 days
M(30402457)	Dec 15 2005	+300 days
M(32582657)	Sep  4 2006	+263 days
M(43112609)	Aug 23 2008	+719 days
M(37156667)	Sep  6 2008	 +14 days
M(42643801)	Jun  4 2009	+271 days
M(57885161)	Jan 25 2013	+1331 days
M(74207281)	Jan  7 2016	+1077 days
M(77232917)	Dec 26 2017	+719 days
M(82589933)	Dec  7 2018	+346 days
------------------------------------------
			       +8059 days (=503.7 days average)

[kriesel]

Originally Posted by ATH

today July 30th we reached the previous record gap of 1331 days, so tomorrow will be a new record This is not a record we want to beat.

It's a record I expect to be exceeded repeatedly going forward. I made an extrapolation in late 2018 of completion of first testing to exponent 1G by around 2170 (152 years from 2018, ~148 years from now). There's an estimated 6 Mersenne primes to be found in that span. So that suggests an average around 25 years between future Mp discoveries below 1G exponent. In the few years since that extrapolation, with Ben Delo's substantial help we made faster progress than the extrapolation for a while, ~1.4x, but not drastically so.
I expect multiyear to decades intervals between Mp discoveries going forward, at least until one or more of the following occur to change the effort scaling, effort applied, or discovery rate:
1) quantum computing or graphene computing become readily available and widely employed in the search
2) we hit another lucky stretch with very closely spaced Mp
3) many times more participants deliberately join the project (or, mprime/prime95 service versions are implemented as standard installed apps or the idle loop in future releases of some major operating systems, which seems unlikely) The number of computer owners globally is about 10⁶ times as many as the number of GIMPS users seen active in the past 30 days.
4) some number theoretical advance is found, and significantly speeds the testing process further. I think the odds of this are low. Some parts of GIMPS are approaching known theoretical limits. (fft multiplication or squaring efficiency, nearly complete elimination of double-check effort by PRP/proof, and reduction in erroneous primality test by GEC)
5) ASICs such as Google TPUs become useful in the search and heavily utilized
6) some other transformative change not anticipated

Ben Delo's contribution rate has declined. User count has declined.
I hope to see at least one more Mp discovered in my lifetime. If one is located near 1.47576 x 82.6M ~122M, reaching and finding it may only take a few more years from now. I'd be happy to be shown unduly pessimistic in regard to time required to find Mp52* or Mp53*.
Some very rough schedule estimates for discovering future Mp, assuming 1.47576:1 spacing and ~6M/year first test advance (optimistic compared to first half of 2022):

Mp*#   p~   year  delta years from ~M(#-1)
52    122M  2024     6
53    180M  2034    10
54    265M  2048    14
55    392M  2069    21
56    578M  2100    31
57    853M  2146    46

[ATH]

Originally Posted by kriesel

It's a record I expect to be exceeded repeatedly going forward. I made an extrapolation in late 2018 of completion of first testing to exponent 1G by around 2170 (152 years from 2018, ~148 years from now). There's an estimated 6 Mersenne primes to be found in that span. So that suggests an average around 25 years between future Mp discoveries below 1G exponent.

This estimate is not proven and we had 13 between 10⁷ and 10⁸, so who knows how many there will be.