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


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


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


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:


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:


... 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.

Tuesday, March 6

Facts first

This local red-light camera setup became operational on 20 October 2017. The first photo shows the scene on the northeast corner of Lawrence Ave. W. and Weston Rd. A second picture shows it from the other direction:

A reporter for WestonWeb (an acquaintance to whom I still owe a couple of beers) seemed to be unaware of its existence when he suggested that "there are only 77 red light cameras in the whole of Toronto and only one remotely close to our area (at Keele and Lawrence)". His erroneous 77 was apparently based on this City of Toronto website:

I attempted to get him to acknowledge that the number was small but he didn't think that that was too important: "... whatever the actual number and location is, it’s inadequate" and then doubled down on the number: "I believe that unmarked red light cameras should be at many more intersections than the current 77."

Banana, banana, banana! It wasn't too difficult to extract the actual apple from the basket. The supervisor of Toronto's Red Light Camera Operations assured me that — as of today — there are 124 red-light cameras in operation.

Tuesday, February 27


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

Saturday, February 24

The long compute

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

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

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

L(40945,328)  <103013>     Norbert Schneider  Dec 2014
L(41507,322)  <104094>     Norbert Schneider  Dec 2014
L(222748,3)   <106278>     Anatoly Selevich   Dec 2010
L(45405,286)  <111532>     Norbert Schneider  Apr 2015
L(48694,317)  <121787>     Norbert Schneider  Aug 2015
L(234178,9)   <223463>     Anatoly Selevich   Jul 2011
L(255426,11)  <265999>     Serge Batalov      May 2014
L(314738,9)   <300337>     Anatoly Selevich   Feb 2011
L(328574,15)  <386434>     Serge Batalov      May 2014

Wednesday, January 24


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

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

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

Saturday, January 13

Ice jam

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

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

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

Here are a couple more views from Raymore Park:

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

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

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