Snow Skunk

Yesterday morning, while still snowing, a skunk moved slowly along a tire track in the road.

Perhaps it had been hit or was otherwise distressed but a closeup showed nothing obvious.

Still, the skunk tracks on the road left me wondering.

Ersatz street view

I was born and grew up in Bergheim, a hillside suburb of the town of Neheim, Germany. Neheim itself came to be incorporated into the town of Arnsberg in 1975. It would be nice to be able to view the streets of Bergheim as they are now and compare them to my childhood memories.

Alas, Google's street view has no presence here, nor for that matter in most of Germany. But by a stroke of luck, a recent bicycle ride by Ronny Gängler takes us on a short stretch of Bergheimer Weg that I remember very well and, by inspection, manages to image what I believe to be my childhood house, though technically not my childhood home because that is on the opposite (not visible) part of the semi-detached structure.

Google's sidewalk path

I've been attempting to fix (in both Apple and Google maps) Weston's misplaced businesses for many years now. Because Weston has some five streets that duplicate, in whole or in part, Toronto street names, it has been relatively easy for a Toronto business that does not know its exact address or postal code to end up in Weston. The exact address and postal code can generally be found using Google search and that information can then be fed to the respective platforms. Occasionally work is also required in the pin placement of buildings, as Google's AI will often guess position on the basis of the street number and the actual site may be some distance from the street. This was the case, for instance, with Weston Lions Arena which has an address of Lawrence Ave W but is actually off Hickory Tree Rd.

Less successful has been my attempt to correct a Google Map path artifact:

The green dashed line on the south side of Denison Rd E represents a path running from the old section of Denison Rd E (which although now an adjunct of the new underpass section retains the same name) all the way to Jane St. An examination of the path in Satellite shows that it is just the south sidewalk of the street. On April 24 of this year I let Google know that the path was superfluous by first selecting it and then choosing "road does not exist" from the informant options. Google's use of "road" here is unfortunate because what I selected was the path — not the road. Likely whoever looked at my recommendation decided that Denison Rd E did in fact exist, so my suggestion has not, to date, been acted upon. I have also coloured magenta, in the above, a building outline artifact that I believe was a temporary workers' shelter while the underpass was being built but is obviously no longer there.

Google's own street view imagery archive shows how the street appeared prior to the underpass construction:

looking east, June 2009

During construction of the underpass, there were (at differing times) three paths for pedestrian traffic. The first meandered on the south side of the project from the Ethiopian Orthodox Tewahedo Church to roughly where the Life Community Church is now (but across the street):

path 1, looking west, June 2014

path 1, looking east, August 2014

The second path was what was to become the north sidewalk of the underpass:

path 2, looking east, September 2014

The third path was what was to become the south sidewalk of the underpass:

path 3 (behind the fence, left), looking west, June 2015

It is this third path that seems to have been immortalized by Google Maps. I'll point out that the east section of Denison Rd E was never under construction, so to have the path go all the way to Jane St shows that Google never really understood what was happening here.

Seasoning

October 31

November 22

A glass clearly

In The Annotated Sherlock Holmes, volume II, pages 332-333 (The Adventure of the Empty House), sidenote 15 lends credence to Sherburne Tupper Bigelow's March 1961 observation (Baker Street Journal: Those Five Volumes) on "Maybe you collect yourself, sir."

The old bookseller now mentions three titles. Then he says: "With five volumes you could just fill that gap on the second shelf." Obviously, two of the three titles must have been two-volume editions or one of the three titles must have been a three-volume edition ...

There is nothing obvious about it. Slightly earlier, the book collector was noted as having wedged under his right arm at least a dozen "precious volumes". He says: "Maybe you collect yourself, sir; here's British Birds, and Catullus, and The Holy War — a bargain every one of them. With five volumes you could just fill that gap on the second shelf."

A "volume" is a book that is part of a larger collection. In a very narrow sense, that larger collection may have the same title with individual books numbered, as with The Annotated Sherlock Holmes. However, the word was never (nor is now) always all that focused. The bookseller has at least a dozen volumes under his right arm: it's just another word for books. Connecting the five volumes to the three titles is the intervening "a bargain every one of them." The bookseller points to the three titles, then sweeps that hand over the rest of the books as he delivers the hawkers' mantra. Five volumes of this stack will fill the gap.

Proportionality

In the list of consecutive positive base-ten integers: 1, 2, 3, ...

   the 1111111111st occurrence of 1 is the first 1 in 1111111111,

   the 2222222222nd occurrence of 2 is the second 2 in 2222222222,

   the 3333333333rd occurrence of 3 is the third 3 in 3333333333,

   the 4444444444th occurrence of 4 is the fourth 4 in 4444444444,

   the 5555555555th occurrence of 5 is the fifth 5 in 5555555555,

   the 6666666666th occurrence of 6 is the sixth 6 in 6666666666,

   the 7777777777th occurrence of 7 is the seventh 7 in 7777777777,

   the 8888888888th occurrence of 8 is the eighth 8 in 8888888888, and

   the 9999999999th occurrence of 9 is the ninth 9 in 9999999999.

In other bases, the n-th occurrence of 1 as the first 1 in n is given in A023037 as a base-ten equivalent. Be careful about the offset (base two is the third term). Multiply the number by d for the n-th occurrence of d as the d-th d in n, up to d = base - 1. These n-th occurrence in n assertions are special solutions to the more general problem but they are not the only ones (all base-ten solutions are illustrated here).

On target (revisited)

one

two

three

eleven

twenty-three

twenty-four

twenty-nine

thirty-one

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

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

two billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

six billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eight billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, four hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-six

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-seven

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-eight

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, six hundred twenty-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-one

nine billion, eight hundred eighty-two million, five hundred thirty-six thousand, one hundred seventy-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred eighty-six million, seven hundred ninety-nine thousand, nine hundred ninety-eight

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred eighty-nine

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-one

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-two

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-three

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-four

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-five

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-six

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-seven

nine billion, nine hundred ninety-eight million, three hundred ninety-nine thousand, nine hundred ninety-eight

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-seven

ten billion, two hundred forty-two million, four hundred sixty-three thousand, eight hundred twenty-eight

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven

ten billion, five hundred twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

twelve billion, two hundred twenty-two million, two hundred twenty-two thousand, two hundred twenty-two

sixteen billion, six hundred sixty-six million, six hundred sixty-six thousand, six hundred sixty-six

eighteen billion, eight hundred eighty-eight million, eight hundred eighty-eight thousand, eight hundred eighty-eight

...

Breakout

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:

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.

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 for the moment the fifth largest known such prime:

386434  (328574,15)      Sergey Batalov       May 2014
300337  (314738,9)       Anatoly Selevich     Feb 2011
265999  (255426,11)      Sergey 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.