Thursday, October 01, 2020

On target (revisited)

one hundred eight
one hundred nine
one hundred ninety-eight
one hundred ninety-nine
two hundred forty
two hundred forty-one
two hundred forty-three
two hundred forty-four
two hundred forty-five
two hundred forty-six
two hundred forty-seven
two hundred forty-eight
two hundred forty-nine
two hundred fifty
two hundred fifty-one
four hundred fifty-three
four hundred fifty-four
five hundred fifty-nine
one thousand, one hundred seventy-four
one thousand, seven hundred sixteen
five thousand, five hundred fifty-six
five thousand, five hundred fifty-seven
six thousand, nine hundred fifty-six
six thousand, nine hundred fifty-seven
fifteen thousand, seven hundred fifty-six
seventeen thousand, one hundred fifty-five
twenty-four thousand, nine hundred ninety-eight
twenty-four thousand, nine hundred ninety-nine
forty-three thousand, five hundred sixty-eight
forty-three thousand, five hundred sixty-nine
seven hundred thirty-five thousand, seven hundred fifty-nine
one million, one hundred five thousand, eight hundred five
one million, one hundred five thousand, eight hundred six
one million, one hundred five thousand, eight hundred seven
one million, one hundred seven thousand, seven hundred eighty-four
one million, one hundred seven thousand, seven hundred eighty-five
one million, five hundred eighty-four thousand, five hundred three
one million, five hundred eighty-four thousand, five hundred four
one million, seven hundred seven thousand, nine hundred forty
one million, seven hundred seven thousand, nine hundred forty-one
one million, nine hundred twenty-one thousand, five hundred sixty-six
one million, nine hundred twenty-one thousand, five hundred sixty-seven
two million, two hundred seventeen thousand, seven hundred nine
two million, two hundred seventeen thousand, seven hundred ten
two million, five hundred sixty-eight thousand, seven hundred one
two million, five hundred sixty-eight thousand, seven hundred two
three million, five hundred forty-one thousand, two hundred seventy-nine
three million, five hundred forty-one thousand, two hundred eighty
three million, five hundred forty-one thousand, two hundred eighty-one
three million, five hundred forty-one thousand, two hundred eighty-two
three million, five hundred forty-one thousand, two hundred eighty-three
three million, five hundred forty-one thousand, two hundred eighty-four
three million, five hundred forty-one thousand, two hundred eighty-five
four million, five hundred twelve thousand, five hundred ninety-six
four million, five hundred twelve thousand, five hundred ninety-seven
twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven
twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

Monday, September 28, 2020


The red polygon atop a Google map (click on it for a better view) represents the 15 hectare confines of my time in south Weston / north Mount Dennis since March 20, with the sole exception of a trip to the vet on June 10. Yesterday, having finally received some cloth masks via Amazon, I decided to break out and walked to the yellow open-circle spot to the northwest, where I took this photo of a condo construction site:

Monday, September 21, 2020

Root words (revisited)

A few days ago, Futility Closet highlighted a David Morice "Kickshaws" bit (from a 1997 "Word Ways") dealing with "root words" wherein the number of letters of a specific power of an integer written in English words is equal to that integer. This looked like something I could verify and possibly expand on. In short order, Mathematica came up with over one hundred examples. I trust of course that Mathematica is correctly providing the (American) English equivalents of the large numbers. Also, Mathematica does not ever include the word "and" in its integer wording. The solutions list begins:

3^0 = 1: one
4^1 = 4: four
9^2 = 81: eighty-one
10^2 = 100: one hundred
10^6 = 1000000: one million
10^7 = 10000000: ten million
10^9 = 1000000000: one billion
10^10 = 10000000000: ten billion
34^3 = 39304: thirty-nine thousand, three hundred four
50^7 = 781250000000: seven hundred eighty-one billion, two hundred fifty million
57^5 = 601692057: six hundred one million, six hundred ninety-two thousand, fifty-seven
60^9 = 10077696000000000: ten quadrillion, seventy-seven trillion, six hundred ninety-six billion
66^4 = 18974736: eighteen million, nine hundred seventy-four thousand, seven hundred thirty-six
70^8 = 576480100000000: five hundred seventy-six trillion, four hundred eighty billion, one hundred million
82^6 = 304006671424: three hundred four billion, six million, six hundred seventy-one thousand, four hundred twenty-four
84^5 = 4182119424: four billion, one hundred eighty-two million, one hundred nineteen thousand, four hundred twenty-four
88^5 = 5277319168: five billion, two hundred seventy-seven million, three hundred nineteen thousand, one hundred sixty-eight
88^6 = 464404086784: four hundred sixty-four billion, four hundred four million, eighty-six thousand, seven hundred eighty-four

In addition to the five powers of ten, 88 has both a fifth and a sixth power that qualify. Another integer with multiple powers (59 and 60) is 2326. My final entry (the full list is here):

3411^83 = 169505077051870754738656270298647848404856613833009765867153972878059706815231408155486012246885256492743555141825746171932166403032624146125802858179156124954911215966912803668763183796090700586141654608753614304795644365897817582455004645907480837144225881607352543409279188776048394771808331: one hundred sixty-nine senonagintillion, five hundred five quinonagintillion, seventy-seven quattuornonagintillion, fifty-one trenonagintillion, eight hundred seventy duononagintillion, seven hundred fifty-four unonagintillion, seven hundred thirty-eight nonagintillion, six hundred fifty-six novoctogintillion, two hundred seventy octoctogintillion, two hundred ninety-eight septenoctogintillion, six hundred forty-seven sexoctogintillion, eight hundred forty-eight quintoctogintillion, four hundred four quattuoroctogintillion, eight hundred fifty-six tresoctogintillion, six hundred thirteen duooctogintillion, eight hundred thirty-three unoctogintillion, nine octogintillion, seven hundred sixty-five novemseptuagintillion, eight hundred sixty-seven octoseptuagintillion, one hundred fifty-three septenseptuagintillion, nine hundred seventy-two seseptuagintillion, eight hundred seventy-eight quinseptuagintillion, fifty-nine quattuorseptuagintillion, seven hundred six treseptuagintillion, eight hundred fifteen duoseptuagintillion, two hundred thirty-one unseptuagintillion, four hundred eight septuagintillion, one hundred fifty-five novemsexagintillion, four hundred eighty-six octosexagintillion, twelve septensexagintillion, two hundred forty-six sesexagintillion, eight hundred eighty-five quinsexagintillion, two hundred fifty-six quattuorsexagintillion, four hundred ninety-two tresexagintillion, seven hundred forty-three duosexagintillion, five hundred fifty-five unsexagintillion, one hundred forty-one sexagintillion, eight hundred twenty-five novemquinquagintillion, seven hundred forty-six octoquinquagintillion, one hundred seventy-one septenquinquagintillion, nine hundred thirty-two sexquinquagintillion, one hundred sixty-six quinquinquagintillion, four hundred three quattuorquinquagintillion, thirty-two tresquinquagintillion, six hundred twenty-four duoquinquagintillion, one hundred forty-six unquinquagintillion, one hundred twenty-five quinquagintillion, eight hundred two novemquadragintillion, eight hundred fifty-eight octoquadragintillion, one hundred seventy-nine septenquadragintillion, one hundred fifty-six sexquadragintillion, one hundred twenty-four quinquadragintillion, nine hundred fifty-four quattuorquadragintillion, nine hundred eleven tresquadragintillion, two hundred fifteen duoquadragintillion, nine hundred sixty-six unquadragintillion, nine hundred twelve quadragintillion, eight hundred three novemtrigintillion, six hundred sixty-eight octotrigintillion, seven hundred sixty-three septrigintillion, one hundred eighty-three sextrigintillion, seven hundred ninety-six quintrigintillion, ninety quattuortrigintillion, seven hundred trestrigintillion, five hundred eighty-six duotrigintillion, one hundred forty-one untrigintillion, six hundred fifty-four trigintillion, six hundred eight novemvigintillion, seven hundred fifty-three octovigintillion, six hundred fourteen septenvigintillion, three hundred four sexvigintillion, seven hundred ninety-five quinvigintillion, six hundred forty-four quattuorvigintillion, three hundred sixty-five trevigintillion, eight hundred ninety-seven duovigintillion, eight hundred seventeen unvigintillion, five hundred eighty-two vigintillion, four hundred fifty-five novemdecillion, four octodecillion, six hundred forty-five septendecillion, nine hundred seven sexdecillion, four hundred eighty quindecillion, eight hundred thirty-seven quattuordecillion, one hundred forty-four tredecillion, two hundred twenty-five duodecillion, eight hundred eighty-one undecillion, six hundred seven decillion, three hundred fifty-two nonillion, five hundred forty-three octillion, four hundred nine septillion, two hundred seventy-nine sextillion, one hundred eighty-eight quintillion, seven hundred seventy-six quadrillion, forty-eight trillion, three hundred ninety-four billion, seven hundred seventy-one million, eight hundred eight thousand, three hundred thirty-one

This is likely the largest solution you will see for a while. Mathematica's large-integer naming system is relatively recent and when it was introduced a few years ago it arrived with a bug. Although that "-illions" bug was eventually fixed, the naming of integers 10^306 and larger remains essentially aberrant/unusable. If you are curious about how many times each letter appears in the preceding:

{a,68}, {b,1}, {c,29}, {d,209}, {e,369}, {f,86}, {g,113}, {h,148}, {i,441}, {l,195}, {m,8}, {n,426}, {o,238}, {p,19}, {q,50}, {r,184}, {s,108}, {t,307}, {u,197}, {v,69}, {w,22}, {x,50}, {y,74}.

There are, in addition, 371 spaces, 97 commas, and 70 hyphens. If you are a user of Mathematica, I will alert you to the fact that the hyphen it uses for its integer names is perversely different from the keyboard hyphen that I have used here.

Sunday, September 06, 2020

My largest Leyland prime find

Late yesterday, I found my (to-date) largest Leyland prime: 33845^26604+26604^33845. At 149763 decimal digits, this becomes now the fifth largest known such prime:

386434  (328574,15)      Serge Batalov        May 2014
300337  (314738,9)       Anatoly Selevich     Feb 2011
265999  (255426,11)      Serge Batalov        May 2014
223463  (234178,9)       Anatoly Selevich     Jul 2011
149763   (33845,26604)   Hans Havermann       Sep 2020

The number is the 167th new Leyland prime discovered since I (using xyyxsieve and pfgw) started finding them two months ago. Prior to that I had found 579 new Leyland primes using Mathematica — but that took from 3 October 2015 to 3 July 2020. At my current rate of discovery, I will find my 1000th new Leyland prime on December 9, but that is likely early because I am entering large-number terrain where my finds will be slower in coming. Still, I might have it by the end of the calendar year. We'll see.

Saturday, September 05, 2020

Visually impaired

Going behind the garden shed just prior to sunset, August 25, I had a smallish raccoon approach me atop the neighbour's back fence. It seemed not to notice me until fairly close, whereupon it retreated to that yard's maple tree. I ran in the house to fetch my camera. It was only after looking at the photos just now that I noticed the eyes.

Display bug


I have had insects walk on my iMac display screen before but this one is behind the glass.

Sunday, August 23, 2020

The iMac from hell

I had mentioned in my previous post that I was considering purchasing a new computer "if I can trade in my old kernel-panic-plagued late-2015 one". When I bought this iMac it was pretty much top of the line.

SSDs were expensive and 1 TB was as big as Apple was willing to provide at the time. The standard 8 GB RAM was going to be replaced with 64 GB purchased from Other World Computing (Apple has always been incredibly ungenerous with its memory pricing).

I experienced my first kernel panic on the new iMac on 23 December 2015, 15 days after receipt. There was another one on 9 February 2016. I seem to have been panic-free until May 8 and then more on May 14, May 21, and May 22, when I decided to do a RAM switch. In addition to the kernel panics there was other weirdness happening in my running Mathematica Leyland prime searches. For example, the Mathematica front end would lose its connection with the Mathematica kernel. Mathematica was pretty much the only thing I was running on this machine.

I tried an OS reinstall on June 13 but the situation did not improve. I should of course have brought it back to Apple before its one-year warranty ran out but I was still under the impression that it might be the third party RAM and, besides, Catherine was increasingly anxious about driving and I did not ask her for the ride.

By July 2017 I had boxed the iMac from hell and purchased a new one, the one that I am currently working on. It's been fine but the 2 TB SSD is getting full. A 4 TB SSD in a new iMac should do the trick for a few more years. Plus I can upgrade to 128 GB RAM. The idea of trading in the defective iMac seemed like a good way to reduce the cost.

Phobio is Apple's official Mac Trade In partner for U.S and Canada. When I tried to evaluate nine days ago my late-2015 iMac on Phobio's website (by entering the computer's serial number), I got this:

My device (iMac Core i7 4.0 GHz) did not appear! Selecting one of the Core i5 alternatives instead would of course end up in my iMac being undervalued and this was not fair. So I sent an email (August 14) to Phobio explaining my dilemma. No response. A followup email (August 19) noting the non-response and asking for them to "please let me know" has also not gotten a reply. At this point I am entertaining deep-Apple conspiracy theories about what is going on (and I'm not the sort of person who entertains conspiracy theories).

Strangely, my late-2014 iMac which has never had kernel panics before started to experience them in June 2019. At any rate, I will give up for the moment the idea of trading in the late-2015 machine. I'll likely still purchase a new iMac but am unsure as to when.

Monday, August 10, 2020

A look ahead (reprise)

A year ago today I charted the expected progress on my five-year indexing-the-Leyland-primes project. Having a few days ago finished interval #10, I anticipate intervals #11 to #13 to be done by September, and #14 by October, all thanks to Mark Rodenkirch's xyyxsieve and pfgw.

my current work sheet

As I am no longer burdened by my previous reliance on Mathematica 9, I may even update everything to macOS Catalina one of these days. A new iMac is also being considered (if I can trade in my old kernel-panic-plagued late-2015 one). After I've finished interval #14, I will have tabulated all Leyland primes up to 103013 digits. Interval #15 will start there and continue on to the end of interval #28, reaching 150000 digits:

15  L(40945,328) - L(41507,322)   6612071   105334 (5e9)  6  Aug 28 - Sep 30
16  L(41507,322) - L(222748,3)   13527824   217348 (5e9) 12  Sep  3 - Oct 10 ~
17  L(222748,3)  - L(45405,286)  33460389   536426 (5e9)
18  L(45405,286) - L(48694,317)  69041008
19  L(48694,317) - L(44541,746)  43871809
20  L(44541,746) - L(49205,532)  45659518
21  L(49205,532) - L(49413,580)  18377349   287809 (e10)
22  L(49413,580) - L(49878,755)  54608684
23  L(49878,755) - L(144999,10)  11614904   182243 (e10)
24  L(144999,10) - L(145999,10)   8050111   126465 (e10)
25  L(145999,10) - L(146999,10)   8094919   127396 (e10)
26  L(146999,10) - L(147999,10)   8139747   128441 (e10)
27  L(147999,10) - L(148999,10)   8184494   128015 (e10) 12  Sep 27 - Nov  7 ~
28  L(148999,10) - L(149999,10)   8229120   129812 (e10) 12  Aug 31 - Oct 11 ~

The column after the interval is the number of Leyland numbers between the end-points of that interval, followed by how many of those remain after sieving to the subsequent quantity (in brackets). This is followed by the number of iMac-mini cores working on primality testing and the start and finish dates (~ added if in the future). Initially, in addition to intervals #15 and #16, I will be having a go at interval #28 in order to up my PRPTop production score.

Tuesday, July 21, 2020

An interesting prime sequence

The sequence starts 2, 3, 11, 23, 29, 61, 19, 113, 157, 127, 103, ... These are the red endings of the following (second column) triangular array. The first column numbers are the indices. The third column numbers are the sums of the digits of the second column integers.

There are several constraints imposed in creating our sequence. Each successive term must be a distinct prime. It must be the smallest such prime that allows the following: At index k, take the final k digits of the sequence. The first of those final k digits must not be zero in order that we may have the concatenation of those digits be a k-digit prime (these are our middle numbers). Finally, the sum of those k digits (the third column) must also be a prime.

  1  2  2
  2  23  5
  3  311  5
  4  1123  7
  5  12329  17
  6  232961  23
  7  3296119  31
  8  96119113  31
  9  119113157  29
 10  9113157127  37
 11  13157127103  31
 12  315712710343  37
 13  5712710343149  47
 14  71271034314989  59
 15  271034314989701  59
 16  7103431498970113  61
 17  10343149897011341  59
 18  343149897011341751  71
 19  3149897011341751379  83
 20  49897011341751379499  101
 21  897011341751379499373  101
 22  9701134175137949937397  109
 23  11341751379499373971223  101
 24  341751379499373971223293  113
 25  1751379499373971223293601  113
 26  13794993739712232936011471  113
 27  949937397122329360114711303  109
 28  9937397122329360114711303223  103
 29  73971223293601147113032231297  101
 30  712232936011471130322312973547  101
 31  2232936011471130322312973547619  109
 32  32936011471130322312973547619769  127
 33  936011471130322312973547619769683  139
 34  6011471130322312973547619769683433  137
 35  11471130322312973547619769683433503  139
 36  471130322312973547619769683433503563  151
 37  7113032231297354761976968343350356337  157
 38  13032231297354761976968343350356337239  163
 39  322312973547619769683433503563372395333  173
 40  2312973547619769683433503563372395333337  181
 41  12973547619769683433503563372395333337311  181
 42  297354761976968343350356337239533333731147  191
 43  9735476197696834335035633723953333373114771  197
 44  35476197696834335035633723953333373114771673  197
 45  761976968343350356337239533333731147716731889  211
 46  9769683433503563372395333337311477167318895801  211
 47  69683433503563372395333337311477167318895801211  199
 48  683433503563372395333337311477167318895801211409  197
 49  3433503563372395333337311477167318895801211409277  199
 50  33503563372395333337311477167318895801211409277313  199
 51  356337239533333731147716731889580121140927731311003  193
 52  3372395333337311477167318895801211409277313110031607  193
 53  23953333373114771673188958012114092773131100316071109  191
 54  953333373114771673188958012114092773131100316071109281  197
 55  3333731147716731889580121140927731311003160711092811381  193
 56  33731147716731889580121140927731311003160711092811381613  197
 57  311477167318895801211409277313110031607110928113816133361  197
 58  4771673188958012114092773131100316071109281138161333611103  197
 59  71673188958012114092773131100316071109281138161333611103263  197
 60  731889580121140927731311003160711092811381613336111032631283  197
 61  8895801211409277313110031607110928113816133361110326312834153  199
 62  58012114092773131100316071109281138161333611103263128341531697  197
 63  121140927731311003160711092811381613336111032631283415316971039  197
 64  1140927731311003160711092811381613336111032631283415316971039179  211
 65  40927731311003160711092811381613336111032631283415316971039179347  223
 66  277313110031607110928113816133361110326312834153169710391793472971  229
 67  7313110031607110928113816133361110326312834153169710391793472971151  227
 68  13110031607110928113816133361110326312834153169710391793472971151349  233
 69  100316071109281138161333611103263128341531697103917934729711513492351  239
 70  3160711092811381613336111032631283415316971039179347297115134923513943  257
 71  60711092811381613336111032631283415316971039179347297115134923513943541  263
 72  110928113816133361110326312834153169710391793472971151349235139435411459  269
 73  9281138161333611103263128341531697103917934729711513492351394354114592617  283
 74  81138161333611103263128341531697103917934729711513492351394354114592617137  283
 75  138161333611103263128341531697103917934729711513492351394354114592617137937  293

I'm indebted to Éric Angelini for the seed of the idea.

Saturday, July 11, 2020

Rev it up

Under the direction of Mark "rogue" Rodenkirch in mersenneforum, on June 28 I was able to run a probable-prime search/verification program that executes some six times faster than what I can do in Mathematica. Considering that Mathematica is what I have been using for my large-Leyland-prime search for almost five years now, that's an awful lot of wasted time!

In order to more efficiently use this program (OpenPFGW; OS X version, pfgw64), a prime-sieving algorithm (OS X version, xyyxsieve) was recommended as a precursor, in order to trim the available candidate numbers to a much-lesser amount. In Mathematica such a routine is incorporated in its PrimeQ function but PrimeQ will not go beyond its built-in limit. So, on July 2 I ran my first prime sieve.

This is a game changer for me. On Wednesday afternoon, a storm knocked out the power here for longer than I had computer battery-backup.

As a result, my Leyland prime search for interval #9 was interrupted. It was scheduled for completion July 25 but recent experience suggested that it would not actually have finished until a week into August. So I used the heavenly portent to convert my remaining search-space into a format that I could use for xyyxsieve and, subsequently, pfgw64. A lot of ongoing manual fiddling and such was needed for the conversions but this allowed me to get used to the new processes. This morning it completed the searches!

Saturday, June 20, 2020

Lydia, oh Lydia

I was looking through some of my old letters to the editor (mostly to The Globe and Mail) in my online library access to local newspapers when I noticed a 1936 Daily Star hit for my surname "Havermann":
The story had been picked up by a good number of other newspapers. Here is a transcribed version from Whitewright TX (July 9):

Love in his heart, a $13 old-age pension check in his pocket and a 16-year-old bride at his side, Oscar Crawford, 66, Tuesday retired to his eighteen-acre farm for the first honeymoon of his life. Crawford on Sunday received his first State old-age pension check. At noon, he appeared at the Colorado County courthouse with Miss Lydia Havermann, 16, at his side. Clerks declined, however, to issue a marriage license until the father of the bride, William Havermann, appeared and gave his consent. While the father looked on approvingly, County Judge H. P. Hahn read the marriage ceremony. The bride and bridegroom left immediately for the C. W. Ellinger farm, five miles east of Columbus, where the bridegroom is cultivating eighteen acres.

There's a follow-up, datelined UP, Austin TX (July 10). My transcribed version is via the Big Spring Daily Herald:

Oscar Crawford, 66-year-old Colorado county farmer, who recently married Lydia Havermann, 16, may have increased his income by so doing, or he may have lost it entirely. Crawford married Miss Havermann in Columbus on the day that he received his first old age assistance check from the state. The check was for $13. "There is a possibility that his pension check may be increased," Orville S. Carpenter, old age assistance director, said here. "Since he is married he now shares his property with his wife. Thus his income is only half as much." This is the first of such cases to arise since distribution of the checks was begun July 1. Section two of the old age assistance act provides, however, that he can receive assistance only if he "has no wife able to furnish him adequate support." The word adequate is not defined.

The marriage was easy to verify:
Somewhat more difficult was what became of the bride and groom. It had me stymied but my research associate, Marlene Frost, quickly found a large number of relevant documents. First off, we have Oscar's death certificate:
Oscar is shown as having been born on 12 Dec 1869 and dying on 21 Dec 1939. The cause of death appears to be under investigation (the word is "inquest"; thanks Alfy and Cathy for deciphering that). Oscar is "divorced", although (annoyingly) his former partner is referred to only as "widowed". The undertaker is noted as "Wm. Havermann", Lydia's father. William may well have buried Oscar but he was a farm hand, not an undertaker.

Lydia's name appears in the April 1930 census (Fayette County TX) showing her as being 7 years old:
That would have made her only 13 when she married Oscar and 17 when he died. Lydia is not with her family in the April 1940 census. Over the years, the 'r' in this Havermann clan appears to have been dropped. Lydia's parents and three of her four brothers may be found in Sealy Cemetery. Her brother Arnold Havemann (born 24 Jul 1931) is married to a Gloria Lois Abel (born 20 Jul 1937) and they have two children, Charles Arnold Havemann (born 28 Mar 1960) and Tracy Kim Semmler (born 5 Nov 1963).

After considerable digging, Marlene finally unearthed Lydia's birth certificate:
She was named Elzie (born 19 Nov 1922). If one thinks of the name as being a shortened Liz'beth then Lydia shares with that the initial sound. It seems that Elzie may have run away from her marriage to Oscar Crawford and took on an assumed name, ending up in Atoka County, Oklahoma. She had a son, Jimmie Roe Boggs (born 5 Oct 1938). In May 1941, as Katie Belle Boggs, she married a 42-year-old Homer Goodson and bore him eight children. In 1953, her father signed an affidavit changing her name from Elzie to Katie Bell:
Katie Bell died in 1971 and has a final resting place in Farris, Oklahoma. Note that the birth year on her headstone is incorrect.

Tuesday, June 09, 2020

Bill Blair

According to a Canadian Press story today, public safety minister Bill Blair says police misconduct is indefensible. Yet ten years ago, under his leadership, he was defending just such misconduct.

Sunday, June 07, 2020

Late evening family outings

8:03 pm: BBQ/Kumbaya behind 1662 Weston Rd. (distancing? masks? garbage!)

Early morning family outings

5:36 am: baby raccoons climbing our backyard maple tree
5:49 am: nine goslings and proud parents on the Humber river

Thursday, June 04, 2020


Squirrels practice physical distancing by running for the nearest tree. This squirrel quickly learned that the square metal post holding up its stop sign wasn't very arboreal.

Yes, we have no plain bagels

We had another grocery order/delivery yesterday. We're good on most of the essentials (bathroom tissue, paper towels, canned goods, bread) but I'm having difficulty acquiring my plain bagels which have been "out of stock" for a while. In yesterday's order I was prepared to go poppy-seed or even sesame-seed, but they weren't available either. The "everything" bagel was a dollar more than what I'm used to paying for a 6-pack and I wasn't sure from the pictured packaging that it was anything more than poppy and sesame seeds on a plain bagel. So I gambled. Alas, they also contained dried onion and roasted garlic, which are anathema to me. I hope the birds aren't as fussy.

Sunday, May 24, 2020

Time flies

In a recent three-and-a-half-hour video homage to John Conway, one of the participants, Siobhan Roberts, recalls Conway telling her that "time flies like an arrow; fruit flies like a banana." Roberts has written a biography of Conway. Of all the stories she might have used to illustrate the man's wit, this one struck me as odd — since of course the quote wasn't a Conway original.

Sometimes attributed to Groucho Marx, quote investigator "Garson O'Toole" (Gregory F. Sullivan) got it right in 2010 when he traced the core of the quotation to Anthony Oettinger, quoting from Oettinger's September 1966 Scientific American article touching on the subject of grammar by computer (time flies vs. fruit flies; the complication is mentioned as early as 1963 in the Harvard Alumni Bulletin). O'Toole: "By 1982 or before someone juxtaposed the sentences to yield a funny combination which was then assigned to Groucho Marx."

The April 1975 issue of Computers and People already had the juxtapositioning. Lawrence M. Clark wrote "Computer Programs that Understand Ordinary Natural Language" (pages 14-19, 23; page 14 reproduced here, quotation boxed in green). It gets better. Clark's three sentences appeared already in the November 1966 issue of Computers and Automation (different title, same publication). Neil Macdonald (a pseudonym for Edmund Berkeley) wrote a short "Research on Meaning in Programming Languages" (page 10, reproduced here). It is "peach" instead of "banana" but that is not as important as the date.

Saturday, May 23, 2020

Where was Weston's Grammar School?

The Toronto Public Library has photo of a "Weston High School" located on the south side of King Street, west of Elm Street, not far from my parents' former house on Joseph Street. I had never known of a school having been at that location and that piqued my curiosity.

Cruickshank and Nason's "History of Weston" has the school "about a quarter-mile from the Main Street" which would put it closer to Rosemount Ave. than to Elm St. To resolve the location, I looked at a 1913 map that showed structures:

The red brick building in the middle of the above is my candidate for the Grammar School. Look at all the empty lots around it. To further convince myself:

Note that there is a slight forward protrusion of the left part of the building which matches the outline in the map. Finally, a 1924 map shows that the building is no longer there. Looking at the location today, one would see this.

Saturday, May 16, 2020

What's an hour?

This morning a couple of ovenbirds crashed into our kitchen deck-door glass. The one on its side righted itself after a couple of minutes and they both just lay there in obvious shock. Here's a closeup of the upright one:

Catherine looked it up and said to just leave them for a few hours. And indeed, three hours later they were gone. This has happened before, twenty or more years ago (I am guessing), and — somewhat remarkably — the two birds that crashed into the door back then were also ovenbirds!

When I downloaded the photos for this article, I noticed that the time-stamp didn't seem right. I soon realized that I had forgotten to set the daylight-saving time option on my camera back on March 8. Which meant that I had 209 recent photos in my Apple Photos app that needed to be adjusted by one hour. Fortunately, Photos makes this easy. Unfortunately, the app hung ~70% through the process:

So I force-quit and relaunched the app to see what had been accomplished. I could see that some of the photos had added the hour but many had not. Worse, Photos had not gone through the 209 photos sequentially by date, but rather, somewhat haphazardly — a few each day. This is no doubt some sort of optimization procedure that utilizes multiple cores for speed gain (the same thing happens when one is importing photos from the camera). What Apple Photos did not realize is that my Mathematica was already utilizing all four cores on my Mac to calculate a ParallelTable. Perhaps this is why the application crashed.

Anyways, I now had to step through each day's photos and try to determine which ones had been adjusted and which had not. I had the camera's sequential photo numbers and the fact that many of the shots had been taken within minutes of each other to help me in this endeavour. However, for days (and parts of a day) where I had only taken one photo, this did not help. It took me another hour or so to step through my backup and check each photo's original time. Only later did I notice that the get-info on even the modified-time photos still showed the original time stamps.

All in all, it took me longer, I think, to correct all that Exif (when did they stop using all-caps?) data than it took those birds to recover. Moreover, I had to add an hour to the "posted" time on my Echo Beach post because I had originally cheated by back-timing that post to match the then-thought-to-be-correct photo time (to give it a more stream-of consciousness feel).

Thursday, May 14, 2020

Unfasten your safety belt

Two months ago we were strapped in for a year-or-two rollercoaster ride. Many bought into the lockdown as a necessary — but decidedly time-limited — mitigation. Two months at home seems to be about as much as the public can bear. It will be an interesting summer.

Sunday, May 10, 2020


A virus with no sexual drive
Was wantin' to only survive.
But its number then grew
Exponential to two,
Which is odd since it wasn't alive.

with apologies to Randall Munroe

Thursday, May 07, 2020

Ten random-start worlds in Rule 193

I first wrote about Rule 193 here, back in 2014. I decided to revisit the cellular automaton by setting up ten random-start evolutions:

0                           1                           2                           3                           4
5                           6                           7                           8                           9
Clicking on each panel will get you a larger version of that panel. Clicking on the numbers underneath each panel will get you the individual evolutions. The pictures are large: 14714 by 8508 pixels (~20 MB). Your browser will likely shrink each one to fit the browser window, so be sure to click/expand it to see all the glorious detail.

If you think of the downward-moving "lines" as particles, a lot of "physics" happens as the particles collide. How many different particles can you distinguish?

Saturday, May 02, 2020

Only eight

This typewritten table of the decimal expansions of powers of two up to 2^115 dates to when I was fourteen years old. My fascination then with powers of two almost certainly arose as a consequence of having encountered the wheat and chessboard problem. I would have calculated the numbers by hand and the typing layout suggests a slight obsession with presentation decorum, a handicap I've endured to the present day. The digit after the power is the digital root.

I recently had occasion to extend OEIS sequence A305942, the number of decimal powers of two having exactly n digits zero. For any given n, that number is fairly constant (on average a little over 33) but there is significant variation. For n up to 295000, I have found a zero-count as high as 62 and as low as 11. Checking other digits in the same range, I find a high of 65 and a low of 8 (see below). These extrema are outliers of course and statistics might suggest that we can find larger-than-65 and smaller-than-8 examples, if only we chart n large enough. But bear in mind that my current database of n up to 295000 is based on powers-of-two decimal expansions up to 2^10000000. It is not a fast computation.

This graph (click on it to get a better view) shows the number of occurrences (the blue points) of the digit 7 in decimal powers of two from 9100000 to 9240000. The green line represents the value 275923. Although (due to the size of the points and the thickness of the line) it may seem that there are dozens of points on the line, there are in fact only eight (at powers 9141747, 9143624, 9155434, 9163531, 9168298, 9171371, 9174454, and 9190491).

Saturday, April 25, 2020

Fantastic soup

Basic ingredients. Good for what ails ye.

"As a general rule, those who think they know everything about a subject really know very little about it; those who know most feel their lack of knowledge, and are always anxious to learn more."
— The Abstainers' Advocate (1894)

Around 1960, Harold Pullman Coffin cleverly rephrased this as: "The fellow who thinks he knows it all is especially annoying to those of us who do." There's a quotation website that annoyingly confuses this newspaper columnist with creationist Harold Glen Coffin. All of the quotations are Pullman's but the photo and the bulk of the bio is for Glen!

Saturday, April 18, 2020

Bare necessities (reprise)

After my previous attempt at a grocery delivery service I wasn't particularly anxious to go at it again, but it was either that or actually going to the store. I tried the delivery arm of the local Real Canadian Superstore where we usually (i.e., used to) shop. They listed most of the products I was after except bags of big carrots, so I ordered a bag of baby carrots instead. The total for the order was just over $200 and I gave them my credit card info for payment. There's a $4 delivery fee, a 5% service fee, and a suggested 5% tip for my shopper/deliverer. The order arrived a couple of hours ago, a mere 22 hours after I placed the order!

They reduced the 8 cans of green beans that I ordered to 0 cans, 8 cans of lentils to 4, and 8 big containers of yogurt to 3. That's ok. I had ordered 4 two-litre containers of 1% milk. I was brought 4 one-litre containers but they charged me for the two-litre containers. That's not ok. The plain bagels that I ordered were replaced with sesame seed bagels, which might be ok but I won't know till I try one. The 8 PC white-cheddar mac & cheese boxes that I ordered were replaced with KD regular mac & cheese. I thought that was going to be ok but I just made myself a couple of boxes and it's inedible (although I did eat a bit and now I'm feeling queasy). I'm going to have to throw that out and hide the other 6 boxes.

I asked for a refund on the missing milk. I didn't ask for compensation on the mac & cheese because I reported the problem before I tried it. On the plus side, they did deliver all 24 rolls of toilet paper that I ordered!

Update: I went back to the site and asked for a refund on the mac & cheese. After all, it's a business transaction, so why should I shoulder the burden of their mistake? Incredibly, Instacart Support (who seems to be the go-between here) not only refunded the mac & cheese, but also all 4 of the two-litre milks — it should only have been 2 of them. They call it their "customer happiness refund". In return, they hoped I would check their "Good, I'm satisfied" support followup (as opposed to "Bad, I'm unsatisfied"). How could I refuse.

Thursday, April 09, 2020


Somebody sure is pissed!



Even the playground:

Not only has the caution tape been removed but the "Hey there!" sign is gone. Could it be that our neighbourhood exerciser took offence?

The Bottomless Lake

We are falling down,
Down to the bottom
Of a hole in the ground.
Smoke 'em if you got 'em.
I'm so scared, I can hardly breathe.
I may never see my sweetheart again.

John Prine (1946-2020) [on Aimless Love, 1984]

Wednesday, April 08, 2020


It has been six months since I started three of my Mac minis on a Leyland prime search from Leyland #302846980 to Leyland #324766364. That range was divided into 18 parts of mostly 1217740 Leyland numbers each and these parts were distributed across the Mac minis for 18 Mathematica programs to each do a probable prime search. The above three pictures show what I see on those Mac minis, each Mathematica window with its own start date/time, subsequent finds, and at the bottom of each window — determined by manual interruption of each program — how far (the middle of the three numbers) it has come in the search.

The how-far numbers are from 47.8% to 51.5% of their respective search spaces, so clearly I have another six months to go before I am done with this section. I count 35 prime finds of which 34 are new (one is a rediscovery of Norbert Schneider's L(34642,707) that he found in December 2017). Somewhat remarkably, the three Mac minis have run for all this time without interruption. They are hooked up to battery backups and what power failures we have had in the last six months have thankfully all been very brief.

Tuesday, April 07, 2020

Hey there!

"Coronavirus-19 ... can live on metal objects for up to 9 days!"

The name of the virus is "severe acute respiratory syndrome coronavirus 2 " or "SARS-CoV-2" for short (the disease that it causes is COVID-19, which will soon enough become Covid-19 as folk tire of the all caps). Of course viruses are not technically alive, so it is better to say that they are detectable on surfaces for a certain period of time. And in my mind at least, it is not at all clear that detectable equals infective.

Saturday, April 04, 2020


I've been spotting cottontails in the neighbourhood on my morning walks for some weeks now. At least that's what I think they are. In and around Denison Park, along Denison Rd. W., even on Sykes Ave. For decades there's been nary a sign of these critters around here. The coyotes came first. Now rabbits. Makes perfect sense!

Thursday, April 02, 2020

Echo Beach

On Echo Beach, waves make the only sound. On Echo Beach, there's not a soul around.

I am sitting on the back deck in the glorious sunshine, listening to some random tunes in my Join-In-Daily playlist. Could life be any sweeter?

Saturday, March 28, 2020

Thursday, March 26, 2020

Bare necessities

Our last grocery shop was on March 11 at the local Real Canadian Superstore. We only go every three or four weeks so it's important to stock up. Alas, already back then they had no bathroom tissue to sell. So on March 17 I decided to try a grocery delivery service, more specifically Grocery Gateway. There's a minimum $50 order so in addition to $30 worth of "Cashmere" I ordered some canned goods. At checkout, the website wouldn't recognize my credit card information so I opted to pay at the door. I tried subsequently to add my credit card information to the account but there was no way to do that. I'm still waiting for their email response to my query about it. But no matter, when the order arrives I'll tap the credit card so that I won't have to push the buttons.

The scheduled arrival for the order was this morning. I was waiting for it by the front steps. When the order was brought to me I asked about the tap limit. I think he said $50. Damn! My original order was for just over $80. But wait, where's the toilet paper? They didn't include it, which put my total owing under $50. But my tap didn't work for some reason so I had to push the buttons.

Sunday, March 22, 2020

My 500th Leyland prime find

This morning, after a four-and-a-half-day wait, I found my 500th and 501st Leyland primes. The above graph extends what I showed for my 200th find. I have now surpassed Anatoly Selevich's 475 such finds that he computed from January 2003 to July 2011.

Generally, I'm happy with the ongoing search. My 54 dedicated Mac-mini cores have been supplemented in the last few months by 6 cores on my old Mac Pro and 4 on a more recent iMac, which have been working on interval #8 to gain time on the overall computation, the length of which I now realize I did not correctly calculate. More specifically, the three Mac minis that have been working the upper half of interval #14 since early October 2019 were thought to complete their task by July of this year. Instead, they will run for a full year, until October 2020. In effect, that pushes the overall expected spring-2021 completion date to the fall of that year.

Thursday, March 05, 2020

Primes describing digit position

On Monday, Éric Angelini posted this to the Sequence Fanatics Discussion list: S = 11, 41, 61, 83, 113, 101, ... with digits 1, 1, 4, 1, 6, 1, 8, 3, 1, 1, 3, 1, 0, 1, ... at positions 1, 2, 3, 4, ...

11 says: "In position 1 is a 1."
41 says: "In position 4 is a 1."
61 says: "In position 6 is a 1."
83 says: "In position 8 is a 3."
113 says: "In position 11 is a 3."
101 says: "In position 10 is a 1."

Of course, each added prime must be the smallest possible that has not already been used. There's a few early surprises hinting at things to come: 11, 41, 61, 83, 113, 101, 151, 181, 233, 223, 263, 293, 353, 383, 419, 401, 479, 467, 541, 1009, 599, 631, 661, 691, 727, 751, 787, 797, 809, 877, 907, 919, 967, 991, 9001, 1031, ... Term #20 is 1009 because to the end of term #19 we have 53 digits/positions and term #19 says that the next digit (position 54) is a 1. So we need a prime starting with 1 and 1009 is the smallest one that keeps the growing sequence truthful. Term #20 also dictates that in position 100 is a 9. So when we get to term #34 = 991, we now have 99 digits/positions and so the next prime must start with a 9. Why not 997? Because that says that in position 99 is a 7 and we already know that in position 99 is a 1. So we must travel all the way up to 9001 to keep things honest. And that may have repercussions when we get to position 900.

I eventually wrote a Mathematica program that seemed to work extending the sequence. But it was taking a long time finding term #1447. So I had a look at how far it had gotten. Term #1446 was 190901 taking up positions 7006-7011. Perusing the list of prior terms, I saw that positions 7012-7020 and 7022-7024 were already assigned with digits: 191737191?371... Stepping through, 19 is prime, as is 191, but these lie: position 1 is not 9; position 19 is not 1. Continuing, no more primes up to 191737191. Then we can try 1917371911, 1917371913, 1917371917, 1917371919, replacing the ? with 1, 3, 7, 9, but these are not prime either. So we attach the next digit, 3, and replace the ? with 0, 1, 2, 3, ..., 9. We need not go further than 5 because 19173719153, finally, is prime!

So I managed to figure out term #1447 before my program did! In fact, it would not have found it because I had my initial search go up to only 104395301. Here's a graph (click on it) of 1500 terms:


Sunday, March 8: I have run into a second large term at #3868. Term #3867 was 301471 taking up positions 21005-21010. Positions 21011-21020 and 21022-21028 were already assigned with digits: 3713793719?9317373... So #3868 is 371379371929 and #3869 is 31737313.

Monday, March 9: This is now OEIS A333085. It needs a decent Mathematica program!

Wednesday, March 11: I have rewritten my original program to run significantly faster. In fact, the new version has already overtaken the number of terms calculated by the old one. Here's an updated graph.

Monday, March 16: I've reached 12000 terms and primes strictly greater than 700000.

Thursday, March 19: Maximilian Hasler has written PARI/GP code for this sequence which he says computes 10000 terms in a few seconds. Unfortunately, I haven't been able to get it to run.

Tuesday, March 31: I've decided to call it quits at 18000 terms. There was another spike at #16966.