10*9*8+7+6+5+4*321

Happy New Year!

The adventure begins (fingers crossed)

photo/ornament courtesy ScienceSocks

Wildlife

coyote under the streetlight, click to enlarge

Spotted walking from Clouston Avenue through Denison Park on this morning's dogwalk, a furtive coyote has just left the north end of park and is about to walk up Denison Road West. I managed to get a lot closer to a fox that I saw on Sykes Avenue back on November 27:

YYZ

From high above my abode in south Weston, courtesy the latest version of Mathematica, a graphic representation of Canadian and U.S. flight paths yesterday that started or ended at nearby Toronto Pearson International Airport. Zooming in ...

Weston Station

click to enlarge

In the above Toronto transit map, Weston Station is clearly marked on the UP Express and GO Line Service routes, heading northwest out of (downtown) Toronto. As one might expect, it is just south of the extended horizontal line connecting Lawrence West (University, line 1) and Lawrence (Yonge, line 1) subway stations.

Roy Murray wrote an article on WestonWeb.ca about Castlepoint's "Weston Park" proposed development right beside Weston Station, pointing us to a YouTube video by Castlepoint Numa on the project. I watched it on December 2nd and noticed that the location of Weston Station had been moved southeast so as to appear much closer to Eglinton Ave. W. in Mount Dennis. So I decided to make a YouTube comment pointing that out: "There's a transit map at 0:50 to 0:55 where Weston Station pops up at an incorrect location. It's actually a good deal further northwest. Presumably the disingenuous placement is meant to convince the target of this propaganda piece of the station's proximity to the Eglinton Crosstown corridor."

click to enlarge

I noticed today that my comment had been removed and comments turned off.

Pandigital palindromic sums

Two days ago Éric Angelini posted on MathFun:

234567898765432 + 1000000000000000 = 1234567898765432

... noting that (ignoring the arithmetic symbols) the concatenated 47 digits are a palindrome. Éric wondered about smaller palindromic sums where all ten digits are present ("pandigital" in the sense of A171102). Here are the 24 smallest:

12076 + 38354945 = 38367021
12076 + 83854945 = 83867021
12376 + 80854945 = 80867321
12876 + 30354945 = 30367821
13086 + 27254945 = 27268031
13086 + 72754945 = 72768031
13286 + 70754945 = 70768231
13786 + 20254945 = 20268731
21067 + 38354945 = 38376012
21067 + 83854945 = 83876012
21367 + 80854945 = 80876312
21867 + 30354945 = 30376812
23087 + 16154945 = 16178032
23087 + 61654945 = 61678032
23187 + 60654945 = 60678132
23687 + 10154945 = 10178632
31068 + 27254945 = 27286013
31068 + 72754945 = 72786013
31268 + 70754945 = 70786213
31768 + 20254945 = 20286713
32078 + 16154945 = 16187023
32078 + 61654945 = 61687023
32178 + 60654945 = 60687123
32678 + 10154945 = 10187623

What a difference five months make

Five months ago, I showcased the then-ten-largest-known Leyland primes. Here's an update:

500973  (100207,99856)   Gabor Levai          Nov 2021
500702  (100263,98600)   Gabor Levai          Sep 2021
386642   (81650,54369)   Yusuf AttarBashi     Jun 2021
386561   (80565,62824)   Hans Havermann       Aug 2021
386548   (83747,41272)   Hans Havermann       Aug 2021
386434  (328574,15)      Sergey Batalov       May 2014
302858   (84181,3960)    Hans Havermann       Nov 2021
301815   (83278,4209)    Hans Havermann       Sep 2021
301763   (64431,48250)   Hans Havermann       Sep 2021
301716   (67237,30714)   Hans Havermann       Sep 2021

The first column gives the number of decimal digits. The Leyland number of an (x,y) pair is x^y + y^x. Nine of these ten entries are new. Of course my list of all known Leyland primes is always up-to-date.

Johnny Strides does Weston

What's so special about 6987952709035199279905674?

Éric Angelini came up with this yesterday:

abcdefghij... (letters are distinct digits):
a is divisible by 1 with remainder 0
ab is divisible by 2 with remainder 1
abc is divisible by 3 with remainder 2
abcd is divisible by 4 with remainder 3
etc.

The largest such integer would be 95674103:

9/1 leaves a remainder of 0

95/2 leaves a remainder of 1

956/3 leaves a remainder of 2

9567/4 leaves a remainder of 3

95674/5 leaves a remainder of 4

956741/6 leaves a remainder of 5

9567410/7 leaves a remainder of 6

95674103/8 leaves a remainder of 7

Much more interesting is if we dispense with the "distinct digits" limitation. Let us allow any digit. The largest integer is now 6987952709035199279905674:

6/1 leaves a remainder of 0

69/2 leaves a remainder of 1

698/3 leaves a remainder of 2

6987/4 leaves a remainder of 3

69879/5 leaves a remainder of 4

698795/6 leaves a remainder of 5

6987952/7 leaves a remainder of 6

69879527/8 leaves a remainder of 7

698795270/9 leaves a remainder of 8

6987952709/10 leaves a remainder of 9

69879527090/11 leaves a remainder of 10

698795270903/12 leaves a remainder of 11

6987952709035/13 leaves a remainder of 12

69879527090351/14 leaves a remainder of 13

698795270903519/15 leaves a remainder of 14

6987952709035199/16 leaves a remainder of 15

69879527090351992/17 leaves a remainder of 16

698795270903519927/18 leaves a remainder of 17

6987952709035199279/19 leaves a remainder of 18

69879527090351992799/20 leaves a remainder of 19

698795270903519927990/21 leaves a remainder of 20

6987952709035199279905/22 leaves a remainder of 21

69879527090351992799056/23 leaves a remainder of 22

698795270903519927990567/24 leaves a remainder of 23

6987952709035199279905674/25 leaves a remainder of 24

Oaklands

click to enlarge

This is my first-year high school class of 1964/1965. Last Saturday, one of these teenagers hosted a high school reunion celebration in his residence at 50 Yorkville Avenue. If one were to look roughly northwest from the top of that building, they might see something like this:

click to enlarge

I have marked the De La Salle college location in yellow. This Apple maps representation is ~2018 because construction on the townhouses (just west of Del's fussball pitch) on Avenue Road is in its starting phase. Additionally, there are now townhouses to the south of the field on Oaklands Avenue. The city of Toronto did a detailed heritage property background report in 2015. Here's a closer bird's-eye view:

click to enlarge

There is a worthwhile article on De La Salle, written in 1980 by William Stephenson, called All Things Right & Dutiful. Somewhat surprisingly for such fare, it contains a paragraph on the once-upon-a-time corporal punishment meted out to some of the students.

I was unfortunately unable to attend this 52nd anniversary (counting from our graduating year) get-together. Many of my reasons noted here were of course tongue-in-cheek.