Air show

Looking southeast from my back deck, I'm guessing these are the U.S. Navy Blue Angels.

A million-digit Leyland prime (encore)

Hot on the heels of my million-digit Leyland prime find, I have come across another one:

197180^119151+1*119151^197180 is 3-PRP!

This one was a little unexpected because it is so close in size to my other one, a mere five decimal digits less. The current top-five Leyland prime leaderboard now looks like this (the first column is the number of digits):

1717671  (1343238,19)      Ryan Propper    May 2023

1433792   (300102,59935)   Ryan Propper    May 2023

1268947  (1139148,13)      Ryan Propper    Jul 2023

1000910   (191319,170462)  Hans Havermann  Jul 2023

1000905   (197180,119151)  Hans Havermann  Aug 2023

833719/265381

Based on the simple continued fraction of π, its convergents (rational approximations) are: 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, ...

Prime numerators are at position 1, 5, 9, ... Prime denominators are at position 2, 4, 9, ... The ninth convergent therefore has both prime numerator and prime denominator, noted ~2003 in the OEIS. It seems unlikely that we will ever see another such.

I thought it might be useful to have here a listing of the positions of prime numerators (p/) and prime denominators (/p) so as to better assess the rarity of their confluence:

1/
/2
/4
5/
9/
/9
11/
16/
/33
87/
230/
334/
594/
/595
840/
853/
/1127
1149/
/2003
2726/
/3611
3788/
/4356
/6926
7442/
8751/
/25333
/27652
/32395
/37722
42038/
/114199
143753/
...

A million-digit Leyland prime (I got lucky)

Back in April, I wished myself "better luck" in my current million-digit Leyland prime search. I've won that lottery, so to speak:

191319^170462+1*170462^191319 is 3-PRP!

The number has 1000910 decimal digits. The current top-five Leyland prime leaderboard now looks like this (the first column is the number of digits):

1717671  (1343238,19)      Ryan Propper    May 2023
1433792   (300102,59935)   Ryan Propper    May 2023
1268947  (1139148,13)      Ryan Propper    Jul 2023
1000910   (191319,170462)  Hans Havermann  Jul 2023
1000175   (218767,37314)   Gabor Levai     Mar 2023

Trouble above

I noticed mid-June a stain on the ceiling of my main-floor library room, likely a consequence of the 30 mm precipitation received on June 12. It necessitated a new roof:

click to enlarge

Busy spell

kitchen @ 3:45 am: click to examine the chaos in greater detail

Goodbye landline

My brand new iPhone 13 arrived yesterday. I wasted little time in trudging up to Freedom Mobile (1924 Weston Rd.) in the heat — twice (I didn't bring any identification the first time)! So now I have a cellphone number. This meant that I could lose the "home" phone which I had been using only as a two-factor authentication device. When I called Bell, they dragged out the cancellation process and finally offered me the landline (I'm paying $60 per month) for something closer to $10 per month (if I heard correctly). "Just cancel the fucking phone," I blurted out uncharacteristically.

Shoe change

21 December 2014

14 June 2023

Little free library

Ross and Laurie (beside Denison Park at Lippincott) have set up this very nice "little free library" box in a corner of their place. I should be able to donate a few books from my own mostly-reference library. Not everyone thinks that these things are a good idea.

Agent orange

This late-evening marauder comes as close to being "orange" (especially in the tail) as one might deem possible. Close inspection suggests that the effect might just be a whitened mixture of "brown". A search finds that the descriptive word for it is "erythrism". This particular raccoon paced our back- and side-yards for a half-hour or so before finally climbing the fence into a neighbour's yard. This included ventures onto the back deck and peering into the kitchen door. When I went out in order to scare it away, it was somewhat unconcerned by my approach — perhaps even attracted to it — as though it was tame. This leads me to suppose that it might be suffering from distemper.

Update: Two days later it was on its way to be euthanized...