Saturday, July 14

Digit sums of powers of integers


Cliff Pickover's Thursday twitter "shiver" explained: "666 is equal to the sum of the digits of its 47th power." That is really only half the story. 666 is also equal to the sum of the digits of its 51st power. I've created a cheat sheet to easily find exponent solutions for integers up to 20034 (note that not all integers have a solution). With it (and nothing else) you can solve this problem:

The sum of the digits of n^911 is n. What is the sum of the digits of n^913?

Sunday, July 8

(10^10002-1)/9-10^2872

This is what (10^10002-1)/9-10^2872 looks like:

111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111101111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

A zero (I've made it bold to make it easier to spot) is sandwiched between 7129 ones to the left and 2872 ones to the right. A number composed only of ones is called a repunit, so this would be a near-repunit.

The number is prime of course but that's not what makes it special. It would be relatively easy to come up with other near-repdigit primes of this (or even much larger) size. What makes it special is that no other distinct permutation of its digits is prime.

Think of it this way: Consider the repunit composed of 10002 ones. There are 10002 different integers where one of the ones is replaced with a zero. When the initial one is replaced with a zero, that zero disappears and the integer becomes a repunit composed of 10001 ones. Every other substitution (having the zero in a different place) maintains the 10002-digit length. We test all 10002 thus-permuted integers for primality (which may take a while) and discover that they are all composite except for one, i.e. the provided example.

There is a sequence of such primes (A039986) in the OEIS and I'm currently trying to extend the number of its known initial terms.

Friday, May 25

A pre-European aboriginal settlement in Weston

In 1937, Frederick David Cruikshank and Joseph Nason published their "History of Weston" in which the authors gave short shrift to the place's aboriginal inhabitants. On page 2 of the first chapter (The Beginnings) they wrote:

"The Humber in this district was a favourite haunt of the Ojibway tribes and it is said that they visited the old hunting ground as late as 1841. There were at least two Indian burial places close to Weston, one just south of the village and the other about a mile to the north. In 1825, John Paul, the well-known cooper at The Humber, saw the internment of an Indian warrior and his accoutrements."

I was curious about that burial place "just south of the village". The context is clearly the original village of Farr's Mills which was on the flood plain of the Etobicoke side of the Humber river, roughly where the Weston Golf and Country Club is now. Just south of there is St. Philip's Anglican Church with its small cemetery. Further south still is Riverside Cemetery in the Humber Heights community, which I have argued strongly (nine months ago) is historically part of the greater Weston community. Could either of these two cemeteries have commingled with an earlier native burial site?

I don't know. But if we take Cruikshank and Nason at their word, there was an aboriginal burial place in this area and one doesn't have a burial place without an accompanying settlement. So, what evidence is there for a native settlement in Humber Heights? The Royal Ontario Museum has a number of native artifacts "from the Weston area". It is likely phrased that way because they weren't actually found within Weston, as the town came to be incorporated strictly on the east side of the river. Rather, they were from near there. There are records of an A.M. Kennedy, Esq., of Weston, having gifted a large number of native artifacts to the museum in 1914 and at least one of these came from a site "north of Weston" so I can't assume that any of his artifacts might have come from Humber Heights. In fact, I think it likely that all of Kennedy's finds might have come from a known Iroquois settlement north of Weston.

There does exist a record of a Weston archaeological site which was supposed to be a native "village", as "confirmed through surface sample and comments of local inhabitants" (Victor Konrad, 1972). Strangely, Konrad placed this site at the Humber Place apartments, in Weston, not in Humber Heights across the river. The Humber Place apartments were built in 1969, so Konrad was unable to carry out his own survey. If that was the case, how did he get a surface sample? He didn't. Apparently Victor Konrad learned of the site from a "Father Meighan" (from Upper Canada College). Father Meighan's name is attached to a number of Toronto area archaeological sites surveyed in 1950. It's possible that Humber Heights was one of those and that he is the one that did the surface sampling and talking to the locals. Konrad simply passed on that information. There is more. Konrad gave the site's location:

Cultural Matrix: 3-6 acres
Latitude: 434150
Longitude: 793140
Easting: 1930
Northing: 3925

Clearly Konrad did not come up with these numbers on his own as they place the village in Humber Heights. He must have transcribed them from Father Meighan's report! But if Meighan noted the location in Humber Heights, how (and why) did Konrad end up at the Humber Place apartments? It's a bit of a headscratcher. One possibility is that Meighan referred to the locale as "Weston" and a confused Konrad, aware only that the town of Weston had in 1967 been absorbed into the Borough of York, simply assumed (without referring to the site coordinates) that Meighan had meant the site to be east of the river and thus described it as so being there ("east of present arena").

There is a way to confirm or refute this hypothesis: Find Father Meighan's report on the site! Unfortunately this may not be possible. Apart from Father Meighan's name being associated with some Toronto archaeological surveys, I have been unable so far to find him in any other context. I haven't even been able to figure out his first name. And as this man has seemingly vanished from the face of history, so too has his report.

Nowadays, most folk peripherally acquainted with the matter will attest to the aboriginal "village" archaeological site as being at the Humber Place apartments and cite the nearby (less than 200 meters away on the other side of the Toronto Community Housing high-rise on Bellevue Crescent) "Weston bones" archaeological site as a native burial ground for it.

I must remind these people that the fourteen-or-so skeletons dug up there in 1911 constitute a site of unknown date and culture. The Toronto Historical Association suggests that they "may be French, since Indians rarely buried their dead in high-traffic areas" (Weston Road, the Carrying-Place trail, is 60 meters away). They could just as easily be English "who fought in the rebellion of 1837 or the war of 1812", as one Westonian suggested in a newspaper report at the time. We do know that Dr. Rowland B. Orr (Superintendent of the Provincial Museum), who visited that site more than three days after the bones had already been dug up with many of the skulls — by then — pilfered, almost certainly mischaracterized the skeletons to be "Indians" based on a supposed "arrangement" of the bones that is directly contradicted by eyewitness testimony.

Significantly perhaps, in the 1911 newspaper articles none of the Westonian gossip was to the effect of there having once been an Indian village close by!

Wednesday, May 16

Weston's archaeological sites

There are two archaeological sites in Weston and this morning I visited the Toronto Reference Library to see if I could document where exactly they might be. The book was Victor Konrad's 1973 "The archaeological resources of the Toronto planning area". I was more than a little disappointed with what little detail I found.


The two open circles represent Weston's two sites in the then- Borough of York, the upper-left one called "Scarlett Rd." (AkGv-5) and the lower-right one called "Bellevue Cr." (AkGv-6). I dealt with the Bellevue Crescent "Weston bones" site here, so we know the exact location of it. The circle in Konrad's map has it too far to the right by some 250 meters. I wonder if the Scarlett Rd. circle is similarly displaced. At the very least that would bring it closer to Scarlett Road which lies on the other side of the Humber River (the thick wavy line to the left of the circles).

Clearly the circles were not meant to be an accurate placement of the archaeological sites. Konrad lists the Scarlett Rd. site in the Borough of York (a 1967 municipality that absorbed the former town of Weston) but the actual Scarlett Road is not in that borough. Rather, it is in the adjacent municipality of Etobicoke. This discrepancy should be resolvable by a reference that notes the exact location of AkGv-5 but I can't seem to find an original source. It's not just the archaeological sites that have disappeared. Even the information about them is slowly becoming lost!

Tuesday, April 17

A 100031-digit Leyland prime

In February I explained the status of my current Leyland-primes search, hoping therein to have scored my first Leyland prime with more than 100000 decimal digits by this summer.

Today I achieved that milestone with 23406^18781+18781^23406, which contains in its expanded form 100031 decimal digits.

Saturday, April 14

Lost in conversion

I thought I'd give the Netflix reboot of Lost in Space a go just now. To be clear, I'm old enough to have watched the mid-1960s original and I didn't think much of it. Three minutes into the first episode of the remake, the starship is losing altitude:


I've enabled the subtitles in the screen grab so that you might comprehend my objection. I decided not to watch any further.

Tuesday, April 10

Ancestry

I got my ancestry-by-populations chart from 23andMe today:


No real surprises. My father's family is the German component. The Netherlands and Denmark matches just serve to regionalize which part of Germany (northwest) that family lived. Poland is there, of course, because that was my mother's mother's place of birth. We had always thought of my mother's father as Russian and the match for Ukraine is a nominal step in that direction, as one mustn't take these countries literally.

Somewhat more surprising is the 8.7% British and Irish component. It doesn't mean, I'm sure, that I have an ancestor that hails therefrom. Rather, some British and Irish populations share a common ancestry with me.

Wednesday, April 4

California dreamin'

Yesterday morning I noticed a couple of small plastic STUCCO advertising-signs nailed to the hydro utility pole in front of my house and — sure enough — two more on a pole up the street. I've been waging war on these signs for a good number of years! So I surveyed the immediate neighborhood (some six streets) and found two more pairs, on Lippincott St. W. and Edmund Ave. In the afternoon — wearing my high-visibility safety vest, a hammer under my belt and a small ladder on my shoulder — I took them all down.

In my survey I discovered a "medical marijuana" dispensary on the northwest corner of Weston Rd. and Lippincott St. W. (The only time I might find myself at this intersection is for the across-the-street Dairy Queen, which is closed in the winter.)


This is the first such business (of which I am aware) in Weston, albeit the south end thereof. Apparently, it is a concern.

Saturday, March 31

Go fund me

I wrote about our local "reverend" Patrick White of the Chaplain's Office here and here. When I was doing all that research on him I somehow missed the February 2017 fundraiser set up by York St. Peter's, an evangelistic organization based in Woodbridge ON. White's friend, Oliver, is noted as being in "serious condition" after he "was overcome with smoke and suffered 3rd degree burns" fighting a stove fire in White's apartment — which Oliver had been assigned to "watch" while White was out of town. Oliver, I guess, was augmenting his watching with a little cooking.

Another casualty of the fire was Patrick White's parakeet, Joey, who came through the ordeal with a bit of smoke inhalation. Joey is mentioned in a photo on the Chaplain's Office website:


Wait a second! Your friend Oliver was seriously hurt in a fire in your apartment and all you can say is that he was recently married ?!

Thursday, March 15

More English number words in base-26 pi

I last worked on this in January 2013. Dropping the final "googol" line from that post's in situ list, but continuing with subsequent finds:

..datnmvbwezlfygflonebugwcjooatbjbskh..
..wzlwdhgbhphxeqsqfourdjjmmikzqwvivabm..
..pceoeiigwyqtmaustwoamxyatxhqxoyzbno..
..ifnuetcfdxrwipodsixyavsnasweexagllo..
..ectasqgcghjmzirrtenfjiphdkdoumqzpjt..
..dwbfgegkqjwkubjvfivewgjlbihzmwgalifm..
..jyhdeevmsxmbxtbininesuzrnrzsxosctuxm..
..efzoqitnhuljsgztzerovlptejdfwretaara..
..hwdvoynebabsyjkdsevenaabgjthpqgapqnrg..
..cgakthjgxsmoldokthreeapdiuhncqcjgsmnt..
..gfaoforthfmdhgkzfiftyxiboudqllqzqlcmh..
..obptjczwqdirtnvmfortyigfxwmybvztjkhbc..
..hubksmltneixkvpheightmgsxrwaqokrdcpnw..
..ttsybfashnuibvlisixtytdchrjcvvzazzivu..
..wltescttnomsztonelevenfyifyclxdcacsfta..
..qpddtdyuitfganmgninetyhecteviwajvcosrv..
..kvvtnvsdypwsumzveightysstqdeajfbyxrrde..
..iuwxlmueqjjzpbcmtwelveefcsszipcnmifmzp..
..ssoshdnbsnembdkgthirtyqwbfxohiadwgtvya..
..fcgcgwvbsyihoavxtwentymbigcbkjfiuezgex..
..edapsvilnlndkwusfifteenzkfkepmwhsauydgc..
..mkczsxsgslvltfxqnineteenpxzlphhysfrfpjwp..
..gmrxvlhvuocxdssaseventyurdrqcdywvowytwb..
..qndvrurhkjaqjqabsixteennwdbnngckhpzduwr..

These are the numbers' first occurrences, at indices 10087, 11324, 13463, 14295, 15276, 64838, 175372, 389247, 786244, 1556763, 2300987, 8879098, 9202330, 9946442, 33027856, 126003234, 126348794, 238426469, 389952198, 536531272, 1709539474, 5624040208, 7690604024, 7869864133, respectively.

I excluded "googol" because it's just a slang name for "ten duotrigintillion". Likewise, I've excluded "hundred", "million", "billion", because in proper English I expect them to be modified by a numerical adjective. Regardless, for the record, here they are:

..dnihymgxncwxfhsdgoogolxvodlfjhbzlnrucr..
..knqqklhgazusrbummillionziaocmauzjxqqrdo..
..bcqtdtvmnmumvmhqhundrednizjohkrhvlzdkpv..
..qgrqqgrlwjuobeyfbillionliuisnrlcobglhiv..

... whose shown first occurrences are at indices 454315613, 942420517, 2169343694, 4359384810, respectively. In December 2000, I created a sequence of numbers: 1, 4, 2, 6, 10, 1, 2, 2, 6, 10, 6, 10, 2, 2, 5, 2, ... representing the (proper) number words in the order in which they occur. I now have a very large number of terms for this sequence. Back then I noted eight consecutive 6s by showing them in blue. In my now-larger list, there are ten consecutive 6s. There are also eleven consecutive 2s, twelve consecutive 1s, and thirteen consecutive 10s.