My 400th Leyland prime find

This latest addition to my table came in just after midnight. It was (at the time) asterisked because I had been unable to submit it (as well as seven previous others) to PRPtop, where submissions were timing out on the "captcha" (I'm not a robot) requirement. This was annoying to me for a number of reasons. I had to worry about Norbert Schneider rediscovering one of these already-found primes and I also use the PRPtop numbers to double-check my total counts.

Update: Finally, on November 25, the submissions started to work again.

Revenant gems

Éric Angelini's revenant numbers became OEIS sequence A328095. In Éric's article he mentions this "gem" found by Jean-Marc Falcoz, 62227496:

6*2*2*2*7*4*9*6 = 72576 * 62227496 = 4516222749696

Subsequently, Chai Wah Wu found 6886826188:

6*8*8*6*8*2*6*1*8*8 = 14155776 * 6886826188 = 97488368868261888

Allowing zeros in the product in any but the units position, I suggested 3691262781:

3*6*9*1*2*6*2*7*8*1 = 217728 * 3691262781 = 803691262781568

Whereupon Giovanni Resta came up with the significantly larger 774773248793:

7*7*4*7*7*3*2*4*8*7*9*3 = 348509952 * 774773248793 = 270016187747732487936

Two more

When on August 1st I showed off my Leyland-prime-find farm (six of seven Mac minis, here), I noted in the final paragraph that I was desirous of two more Mac minis but was low on cash. On September 1st I received an interest payment on a long-term investment that I had forgotten was due, so that by ordering the two Mac minis after my mid-September credit card bill arrived I wouldn't have to pay for them until mid-November, by which time the unexpected interest payment, combined with my upcoming monthly pension and old-age benefits, would be enough to pay for the machines! They arrived yesterday:

I moved the printer from my rolltop desk's back ledge to make room. The task on Mac mini #7 was moved to one of my iMacs, thus freeing it, along with Mac mini #8 and Mac mini #9, to contribute to the Leyland prime search. Thus, I have added 18 cores to the 36 already in the farm. This should take 8 months off my projected two-year search schedule. As a test run, I've started the three additional Mac minis to do half of interval #2. Then I'll set them to run half of interval #14 (which will take 8 months), after which they'll tackle the other half of interval #14 (which will take another 8 months).

J'adoube

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):

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.

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):

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.

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!

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
 6  L(30247,300) - L(40089,82)    8561240  <75828>   38       Feb 23
 7  L(40089,82)  - L(40746,91)   15146841  <78282>   69       May  2
 8  L(40746,91)  - L(32160,329)   5639471  <80390>   27       May 29
 9  L(32160,329) - L(40495,114)  11887307  <82129>   57       Jul 25
10  L(39070,143) - L(91382,9)    15717090  <85712>   78       Oct 11
11  L(91382,9)   - L(35829,302)   8886580  <88031>   46       Nov 26
12  L(35829,302) - L(37738,243)   6370928  <89444>   33       Dec 29
13  L(37738,243) - L(38030,249)   6038222  <90579>   32  2021 Jan 30
14  L(37614,265) - L(40210,287)  43838597  <95032>  241       Sep 28

#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.

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.