Area

The blue mesh consists of unit squares. What is the area of the black rectangle?

The long compute

Around the middle of 2017, I started looking for Leyland primes between L(40210,287) and L(40945,328). L(x,y) = x^y + y^x and we assume x ≥ y > 1. L(40210,287) is Leyland #324766365 and L(40945,328) is Leyland #349812824. So there are 25046458 Leyland numbers between them which I want to check for probable primality. Any given check may take a significant amount of time since we are dealing with numbers that are roughly 100000 decimal digits long.

I haven't been totally committed to the task for the entire period but I may perhaps have spent six months on it. Since I've only covered about one fifth of the territory, I have two years to go! I was going to add some processors to the task but my intended purchase of a new machine has (sadly) been stymied. There was a second issue. My list of sorted consecutive Leyland numbers only went up to #331682621, having been computed with the sole objective of reaching 100000-digit numbers (which it did). I thought I was going to need the new computer to calculate more terms because my indexing computation was limited by available RAM and my current machines can't take any more than 64 GB. Fortunately, I recently discovered that that was sufficient to extend the indexing to L(40945,328).

The good news is that I have so far found eight previously unknown Leyland primes, ranging in size from 98889 to 99659 decimal digits. By this summer I should have scored my first Leyland prime with more than 100000 decimal digits. There are currently only nine known Leyland primes <decimal digits> larger than this:

L(40945,328)  <103013>     Norbert Schneider  Dec 2014

L(41507,322)  <104094>     Norbert Schneider  Dec 2014

L(222748,3)   <106278>     Anatoly Selevich   Dec 2010

L(45405,286)  <111532>     Norbert Schneider  Apr 2015

L(48694,317)  <121787>     Norbert Schneider  Aug 2015

L(234178,9)   <223463>     Anatoly Selevich   Jul 2011

L(255426,11)  <265999>     Serge Batalov      May 2014

L(314738,9)   <300337>     Anatoly Selevich   Feb 2011

L(328574,15)  <386434>     Serge Batalov      May 2014

Pi-primes

3, 31, 314159, and 31415926535897932384626433832795028841, are the first four terms in the "pi-primes" sequence: A005042. That 38-digit fourth term was attributed by Martin Gardner (in 1979) to Robert Baillie and Marvin Wunderlich. By 2000, a larger (fifth) term had yet to be found. That year, Clifford Pickover (under the guise of Dr. Googol) wrote in his "Wonders of Numbers" that there are likely infinitely many terms but "neither humans nor any lifeforms in the vast universe will ever know the next prime... It is simply too large for our computers to find." I wrote about Pickover's gross underestimation of computational progress previously and this serves as another example.

In 2001, Ed T. Prothro calculated that fifth term, composed of 16208 digits. In 2006, Eric Weisstein calculated the sixth and seventh terms, composed of 47577 and 78073 digits, respectively. In 2016, Adrian Bondrescu calculated the eighth term, composed of 613373 digits.

A fine point is that — as of this writing — the fifth to eighth terms are not proven primes, but probable only. That should not deter one from (pragmatically) thinking of them as primes.

Ice jam

Yesterday morning, a mild spell had fractured enough of the local Humber river ice — created over several weeks of bitter cold — to have started a small ice "jam". The ice jumbles reach river bottom and thus prevent water from flowing underneath. Diverted water flows in the Raymore Park floodplain on the other side. Here's how it looked from there:

At noon, the ice backup reached ~300 meters ...

... but an hour later, chunks of new upstream ice were pushing into it:

Here are a couple more views from Raymore Park:

Somewhat surprisingly, the Raymore island beaver was out and about:

By this morning, the river was flowing once again. Some of the ice had become wedged in the narrow channel on the east side of Raymore island:

I believe that this year's ice jam is the first significant one in seven years. My record of the 2010 ice jam (with a link to the 2009 one) is here.

10-9+8*7+654*3-2+1

Happy New Year!

Going for the juggler

It is conjectured that all juggler sequences eventually reach 1. More info: https://t.co/mZOlQgz6KB pic.twitter.com/oo7FxGz1tv

— Cliff Pickover (@pickover) December 5, 2017

Cliff's twitter link ends up here. Eric Weisstein's MathWorld entry is here. Clifford A. Pickover first wrote an article about juggler sequences in November 1990 and challenged its readers to prove that all such sequences fall to 1. He fleshed things out a bit in chapter 40 of his 1991 "Computers and the Imagination". The juggler sequence starting with 37 is:

 0               37

 1              225

 2             3375

 3           196069

 4         86818724

 5             9317

 6           899319

 7        852846071

 8   24906114455136

 9          4990602

10             2233

11           105519

12         34276462

13             5854

14               76

15                8

16                2

17                1

Odd numbers grow the next term; even numbers shrink it. With 37 as our start, it takes 17 steps to get to 1; the largest term (composed of 14 decimal digits) appears here at step 8.

Harry James Smith (27 Jan 1932 - 5 Jun 2010) was an early adherent of the cause and soon found himself looking for record large terms in juggler sequences. In 2008 he found an 89981517-digit term in the sequence starting with 7110201. It's in this spirit that I decided to look for records in either the number of steps needed for a juggler sequence to reach 1 (A007320) or the largest value encountered in a juggler sequence (A094716). Here they are:

# 0           1     0     0            1

# 1           2     1     0            1

# 2           3     6     3            2

# 3           9     7     2            3

# 4          19     9     4            3

# 5          25    11     3            5

# 6          37    17     8           14

# 7          77    19     3            7

# 8         113    16     9           27

# 9         163    43     6           26

#10         173    32    17           82

#11         193    73    47          271

#12        1119    75    49          271

#13        1155    80    24          213

#14        2183    72    32         5929

#15        4065    88    63          386

#16        4229    96    41          114

#17        4649   107    74         1255

#18        7847   131    63         3743

#19       11229   101    54         8201

#20       13325   166    90         1272

#21       15065    66    25        11723

#22       15845   139    43        23889

#23       30817    93    39        45391

#24       34175   193    61         5809

#25       48443   157    60       972463

#26       59739   201    69         5809

#27       78901   258   109       371747

#28      275485   225   148      1909410

#29      636731   263   114       371747

#30     1122603   268   145       209735

#31     1267909   151    99      1952329

#32     1301535   271   122       371747

#33     2263913   298   149       371747

#34     2264915   149    89      2855584

#35     5812827   135    67      7996276

#36     5947165   335   108      3085503

#37     7110201   205   119     89981517

#38    56261531   254    92    105780485

#39    72511173   340   166       621456

#40    78641579   443   275      7222584

#41    92502777   191   117    139486096

#42   125121851   479   203       146173

#43   172376627   262    90    449669621
#44   198424189   484   350      5342028
#45   604398963   327   172    640556693
#46   839327145   224   118   2109464216
#47  1247677915   221   119   3225243807

Five columns: The first is just an identification number. The second column is our starting number. The third column gives the number of steps for that starting number to reach 1. The fourth column gives at which step the starting number reaches a maximum. The fifth column gives the number of decimal digits in that maximum. So you can verify #6 with the example previously provided and note that #37 is as far as Harry J. Smith managed to get. Records are indicated in bold.

To bring the large numbers in a juggler sequence down to a manageable size, one can do a log of the numbers and then a log again. Employing this technique, here's a nice graphic plot of juggler sequences #42 and #43:

What's so special about 619603547897^42?

Last month I presented a record large (338-digit) 35-balanced factorization integer:

  8466772177^34   <338>   k=35

A week later I came across a slightly bigger one:

828145091789^29   <346>   k=36

Today:

619603547897^42   <496>   k=51

The 496-digit integer joined with the 14 digits of its factorization contains exactly 51 each of the ten decimal digits!

Never so soon

The mystifying Göbel's Sequence. 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, pic.twitter.com/hnlH2i5dLx

— Cliff Pickover (@pickover) November 13, 2017

Eight days ago Cliff Pickover tweeted the above item taken from his 2005 "A Passion for Mathematics". The sequence was from Richard K. Guy's 1994 "Unsolved Problems in Number Theory" (2nd ed., E15, a recursion of Göbel), wherein is stated that x(43) of the sequence is not an integer. The OEIS version of the sequence is A003504 and because of an indexing difference, A003504(44) is the first one that is not an integer.

I had a try on my four-year-old Mac Pro with 64 GB RAM and was only able to compute A003504(42) with its 44621322894 decimal digits. That suggested however that when the next iteration of the Mac Pro (capable of 256 GB RAM) comes out in 2018 it should be able to calculate the number. But I know that there are personal computer setups out there right now that enjoy that much RAM, so I shared all this with the folk on my MathFun mail list.

Tomas Rokicki took up the challenge and an hour ago he wrote:

Never say never.

The value has 178,485,291,568 digits.

I was shocked to find this number contains all my private and confidential information, including my phone number, bank card PIN, birth date, and social security number. So I am not giving the entire number in this email. But I can release the following information:

The first 320 digits are

54093091807717826044542731578405024787750317409624868757370340478992429511648638619807912524333411657184465174303568454077330681780797548448509290702448196277551062639897479537453119309492272945338076902365702473821655434625012745648296904194171566061774758927571733261575074080983857558577239883610271418838784612873710

The last 320 digits before the decimal point are

88699191682095825749862787174440182918616379497652579422163481170671348506036049354055890488898970220363641477921737257912963162577020774432667767777793755229346705459056403522101537720330787404894294305808931882192829454219123579914800407010187249364040347536438198239409105844922872410211186280040839110841726300820408

The digits after the decimal point are

558139534883720930232

and then they repeat.

Of course the initial and final digits are pretty easy to calculate; here are the digits after the first 100,000,000,000:

85041615512142567506964927695069635814589786429627680066755429543873436865252126753770549128275487654429237310835737004603362273282618046800405177245117802433985448842221870449898846832719482353455460563011453352498599203254784682189218933685168983969281156766864358251454065067791857528820923836736596870531367771176483

The md5sum of all the digits followed by a single newline is

02e9e02ac6e54b3bf070725121900d17

I've checked the residue of the integral portion of this number against the expected values for hundreds of primes and they all match. For instance, for 1000000007 the residue is 289545313.

What's so special about 8466772177^34?

When I introduced my factorization balancing act last year, I gave the example of a large (13-balanced) factorization integer:

7^85*918679^8   <120>   k=13

The number's 120-digit decimal expansion joined with the 10 digits of its normally-expressed factorization contains altogether exactly 13 each of the digits zero through nine. I suggested that it hadn't been too difficult to obtain — essentially, brute-force searching the product of a prime to a power and another prime to a power. In spite of this number's relatively easy generation, I hadn't found a larger example until recently:

8466772177^34   <338>   k=35

That is, the 338-digit integer joined with the 12 digits of its factorization contains exactly 35 each of the ten decimal digits!

The Reverend

Nine days ago I was made aware of the Reverend Patrick White, his Chaplain's Office, and their "advocacy and outreach work" here in Toronto:

New lease on life spawns second career
Toronto chaplain trainer relishes opportunity 'to see lives change'
Aaron D'Andrea, The York Guardian: 21 September 2017

The two links are online versions of the same story. I noticed that the second one corrected White's age in 2008 from 63 to 65 and the link of The Chaplains [sic] Office of Canada from www.thechaplainsoffice.org to www.chaplainsofficeofcanada.com. The latter domain was updated in July and is good to go until 2022.

Additionally, a Facebook page appears to have been active in 2015 and 2016. An example of the Reverend's outreach work may be found here, and more recently here. Note that in the latter's attached letter he is calling himself Bishop Patrick White — not to be confused with the now-retired bishop of Bermuda whose age is very close to that of our "bishop".

If you would like to hear Rev. Patrick White moralize, there's an hour-long audio sermon here. Although mostly tedious, there are a few gems. Among them is this: "On my father's side I grew up in the funeral work, so we had funeral homes down in a little town called Leamington, Ontario. ... I started at about 14 years old and used to do the embalming upstairs ..." Here's my background check.

Update: As of 2019, Patrick White appears to be plying his trade out of the S.R. Drake Memorial Church in Brantford which he is using to Go Fund Me a Cup of Hope.

Update II: There's very little on him after 2019. He may have moved out of Brantford. A comment on a December 2021 Oshawa Covid-19 news item from a Dr. Rev. Patrick White is very likely him. He writes: "My choice to be vaxinated was due to the fact I visit hospitals senior homes, Hospice, and could not safely continue my work, or be admitted unless I had proof. My faith is in God, but I try to follow man's laws and requirements as well."