Tuesday, August 27, 2019


I have now finished the first assigned interval (#0, here) on my Leyland-prime-find farm. As noted in the final column of my to-do list, I'm a day early. One of my Mac minis finished already two days ago, three finished yesterday, and two today. I was prepared for some computation-length variability but perhaps not quite that much. The mini that finished early was the leftmost of my array, so perhaps (because it was beside the printer) it was able to more efficiently dissipate the heat. On that basis I've decided to stand them all on their heads which directs the heat exhaust upward (instead of back toward the wall):

Saturday, August 24, 2019

Reverse flow

A year or so ago our toilet tank refills began to be accompanied by an annoying whine. Our only available fix was to turn on a faucet in the nearby sink.

There's been construction on the street and last Tuesday an excavator managed to snag the water intake conduit to our house. They attached a baby-blue hose to the city's water supply and connected it to the backyard faucet at the side of the house.

And, sure enough, we again had water in the house. It never occurred to me that this was possible. Presumably, as long as there is no countervailing pressure in the plumbing system, the water can enter the house anywhere — duh! After a few hours they disconnected the hose and fixed the original water intake.

They sorta fixed something else. For a few days the toilet tank refills refrained from whining.

Friday, August 23, 2019

Drive-by shooting

A drive-by shooting happened Wednesday evening some 600 meters (5 blocks) southeast of my home. I went down on Thursday morning to see if there was anything left to photograph. Indeed, the scene was still cordoned off, the bullet-riddled crashed-into-a-tree vehicle on display, and three Toronto news outlets were ready for live on-air reports. Here's the CP24 news team (Jee-Yun Lee holding the microphone):

Thursday, August 22, 2019

The big win

Gerald and Helen (on the right) Phillips of St. Thomas, Ontario, were handed an oversized cheque for $26,000,000,000 today. For some reason the Ontario Lottery and Gaming representative (on the left) and the news organization covering the event suggested that the couple's August 6 win was for considerably less.

Sunday, August 11, 2019

The big fix

In April 2017 I wrote an article on counting t-free ordinals that mentioned (at the end) an issue with Mathematica's then-just-released version 11.1 that incorrectly pluralized newly introduced number names for 10^102 (one "trestrigintillions") all the way up to 10^303 (one "centillions"). A stupid mistake (how did this get past the vetting process?) but an easy (say, ten-minute) fix. "I've alerted Wolfram to the bug," I wrote.

Versions 11.1.1, 11.2, and 11.3 came and went without the fix so I complained again about it (just short of a year after my first alert). Somebody at the Wolfram Technology Group thought it helpful to offer this workaround:

removePluralS[str_String] := StringReplace[str, "illions" -> "illion"]

I did not think to check version 12 when it came out last April. An email from the Wolfram technical support team on August 8 alerted me to the fact that the issue had in fact been resolved. So, two years!

Saturday, August 10, 2019

A look ahead

Now that my Leyland-prime-find farm is running and I am done with the previous-interval search that had still been running on my other Macs, I'm ready to have a look at the future of this endeavour:

 0  L(29934,157) - L(40182,47)    6243569  <66463>   24  2019 Aug 28   Aug 27
 1  L(31870,131) - L(34684,105)  11570518  <68797>   46       Oct 13   Oct 12
 2  L(34684,105) - L(29356,257)   2887602  <70425>   12       Oct 25   Oct 21
 3  L(29356,257) - L(30280,241)   6274269  <71439>   26       Nov 20   Nov 15
 4  L(30280,241) - L(104824,5)    5256668  <72700>   23       Dec 13   Dec  8
 5  L(104824,5)  - L(30247,300)   7747011  <74100>   34  2020 Jan 16   Jan 10
 6  L(30247,300) - L(40089,82)    8561240  <75828>   38       Feb 23   Feb 22
 7  L(40089,82)  - L(40746,91)   15146841  <78282>   69       May  2   May 22
 8  L(40746,91)  - L(32160,329)   5639471  <80390>   27       May 29  (May 22)
 9  L(32160,329) - L(40495,114)  11887307  <82129>   57       Jul 25  [Jul 11]
10  L(39070,143) - L(91382,9)    15717090  <85712>   78       Oct 11  [Aug  6]
11  L(91382,9)   - L(35829,302)   8886580  <88031>   46       Nov 26  [Aug 14]
12  L(35829,302) - L(37738,243)   6370928  <89444>   33       Dec 29  [Aug 20]
13  L(37738,243) - L(38030,249)   6038222  <90579>   32  2021 Jan 30  [Aug 26]
14  L(37614,265) - L(40210,287)  43838597  <95032>  241       Sep 28  [Sep 19, 2020!]

#0 is the interval that I am currently searching. [Remember that L(x,y) = x^y+y^x, x≥y>1.] The quantity after the interval is how many Leyland numbers there are between the interval's end points. After that is a rounded-up average of the base-ten logarithms of all of those Leyland numbers — therefore, the average number of their decimal-digits size.

Following this is a ballpark estimate of how many days the interval search will require. [The 241 days (= 8 months) of the last interval suggests that it should probably be split into parts.] After this is the expected date of completion — assuming of course that all previous days-estimate are accurate, that I am able to start a new interval immediately after completing its predecessor, and that there are no hiccups, such as extended power interruptions or software/hardware failures.

When an interval is completed I will add an actual date after the estimated one so as to provide a sense of how the project is coming along.

Thursday, August 01, 2019

My 300th Leyland prime find

An hour before midnight on Tuesday I found my 300th Leyland prime. It was only this past January when I had found my 200th. Compare that to August 2016 when I found my 100th! The reason for the apparent speedup is that I finished my almost-two-year search of ~100000-digit Leyland numbers in April and started advancing again the Leyland prime index — which had been languishing at #1179 on a 56230-digit prime. In short, since April I have been working on ~60000-digit Leyland numbers and they take much less time to probe (for probable primality) than the 100000-digit variety.

As of today the Leyland prime index (I now keep a backup copy of the document) is up to #1298 but that will be going past #1320 in a week-and-a-half when I finish up my current search interval. Before the end of August I hope to add another thirty (or so) as I start (in a few days) my Leyland-prime-find farm (shown above) to explore the gap between L(29934,157) @ 65733 decimal digits and L(40182,47) @ 67189 decimal digits.

The farm currently consists of six of the pictured 6-core Mac minis (the seventh is working on another problem). For the farm's first coordinated (don't read too much into that word — I'm merely starting the runs at the same time) task, each of the 36 cores will be assigned a bundle of 173430 Leyland numbers to check. The Mac minis are reasonably fast — comparable to my 4.2 GHz 2017 iMac even though the minis are rated at only 3.2 GHz (there's built-in speedup potential if the cores are sufficiently cool). The minis are about 40% faster than my late-2013 6-core Mac Pro which is rated at 3.5 GHz.

While the now-seven Mac minis are a significant upgrade from what I started with 8 months ago, I am thinking that there is room for another two in the array! It'll be a bit of a juggle: I don't want them to come equipped with the next Mac OS (Catalina, which is slated for "the fall") because my running apps are 32-bit and OS 10.15 will only handle 64-bit apps. On the other hand, my savings are drained so it's unlikely that I will be able to make the purchase before the OS release.