Thirty

I turned on the air conditioning yesterday. In spite of that, my room registers at 30º — significantly above the ambient air temperature outside — because there are sixteen computers herein generating heat, 24/7, and there is insufficient air flow from the vents to make much of a difference. The living room is a little better (the ceiling fan helps) but even Bodie has taken to lying on the floor instead of his chair or the bed/couch. I expect that it will be like this most of the summer!

Goslings

My daughter dropped by today with some rescued goslings, wondering if I'd seen at the river any families that might be adoption candidates. Of course I hardly ever see the river these days since my Bodie walks are pre-dawn. I pointed out that historically there had always been such families above the weir in Raymore Park and she proceeded thereto and did in fact find such a family.

Bucket list

1. Wiener schnitzel (The Coffee Mill closed in 2014; Amber closed in 2021.)

2. China Town (Their online ordering system stopped working a few years ago.)

3. KFC (It's a bucket list! Like Domino's Pizza, I've been unable to generate a delivery.)

4. Tim Hortons (I know. How can something so ubiquitous be so difficult to reach.)

5. Harvey's (Overly motivated by a languishing old Ultimate Dining card.)

269 Rexdale Blvd., this morning (after seeing my endodontist)

Original combo, unoriginal price: $10.84

Ovenbird

There's a pair of robins that have been making daily forages around our backyard since at least mid-April, so it made for a pleasant change to see yesterday this ovenbird checking out a small corner-section of the venue.

The new high-rise

Prominent now on my morning walks home from Denison Park is "The Humber", an under-construction condominium at 10 Wilby Crescent — which any reasonably observant map enthusiast will decry as being more properly situated on Hickory Tree Road. The sad reality is that the latter has never properly connected Bellevue Crescent to Wilby, being instead a gated roadway to a couple of other high-rises with access only from the Bellevue side. The yellow structure in the below map outlines the building that used to be #10 (also street-view visible by rotating Google's 2015 no-access part of the road; also the location of my 2020 "breakout" blog):

Even though it still a handful of storeys short of its final height, the new condominium can already be seen from my home:

A million-digit Leyland prime (search start)

Last month I laid out a prognosis for setting up a million-digit Leyland prime search. That endeavour has now started its run!

I sieved my L(999999,10) - L(1000099,10) candidates to 2*10^11 resulting in a 59536-term file. Running the sieve from 10^11 to 2*10^11 was not really necessary. The 12.5 days that it took (on a 10-core machine) netted 1632 composites but a direct primality test would have netted ~50 composites per core in the same amount of time and, at 100 cores, would have resulted in three times the yield. At any rate, the effort was not wasted since nine of my Mac minis are still working on their previous project and are therefore not yet search-ready.

I have now initialized 54 cores on nine different computers to begin the search. In a week I will have added the 54 cores on those nine Mac minis finishing their assignments. So 108 cores on eighteen computers dedicated to the task! I am hoping for completion some time in September. Of course, prime finds (should I be so lucky) could happen at any time.

Leyland numbers with exactly one million digits

Some three months ago, I suggested on Mathematica Stack Exchange that there are 39542 Leyland numbers with exactly one million decimal digits. On my earlier-this-month blog post, I pointed out that I had created a dictionary of 39556982 Leyland-number (x,y) pairs in order of increasing magnitude, starting with the smallest 1000000-digit L(999999,10). The Leyland number L(x,y) = x^y + y^x, x ≥ y > 1. It's easy to cull from this list the first 39542 entries and I present them now here. The final L(190793,174294) is the entry just prior the appearance of the 1000001-digit L(1000000,10).

A million-digit Leyland prime (prognosis)

At the end of my February 21 blog post I suggested that I might to try to find (starting in May) a Leyland prime with one million (or more) decimal digits. I am now in a position to assess what this would entail.

Specifically, I would try to see if there are any (probable) primes with 1000000 or more, but fewer than 1000100, digits. There are 3954322 Leyland numbers in this range but by sieving out ones that are divisible by small primes — say, up to 10^11 — only about 61000 should remain. The sieving can be done in three weeks and only then would I start the search. Each remaining candidate needs about five hours to decide if it was composite, which comes to 35 years overall but, fortunately, I can distribute this across 72 Mac-mini cores, so six months. I will likely add some cores to the job but still, five months!

Considering that I have now spent the last ten months charting 300000-digit Leyland primes, it seems doable. There's a possibility that there are no primes in my working range, in which case I would have to commit to the next-larger range. And so on.

I have created a dictionary of Leyland (x,y) pairs from (999999,10) to (1000999,10), sorted by magnitude and preceded by its Leyland-number index (21588818851 to 21628375832). The text file is more than a gigabyte so I don't see much utility in linking to it. Here is a much abridged version showing the initial-, middle-, and final-100 entries:

21588818851 (999999,10)

21588818852 (333333,1000)

21588818853 (227361,25020)

21588818854 (191945,162114)

21588818855 (307560,1784)

21588818856 (193629,146053)

21588818857 (196205,124941)

21588818858 (192901,152757)

21588818859 (194076,142107)

21588818860 (200954,94680)

21588818861 (201859,89939)

21588818862 (192251,159048)

21588818863 (218510,37709)

21588818864 (195348,131542)

21588818865 (220103,34940)

21588818866 (191577,165893)

21588818867 (190250,180400)

21588818868 (208345,63056)

21588818869 (232052,20388)

21588818870 (200835,95325)

21588818871 (191764,163960)

21588818872 (207423,66231)

21588818873 (197894,113033)

21588818874 (206956,67911)

21588818875 (189903,184434)

21588818876 (193737,145088)

21588818877 (201347,92586)

21588818878 (217470,39658)

21588818879 (236785,16720)

21588818880 (197290,117132)

21588818881 (226287,26252)

21588818882 (196255,124568)

21588818883 (194241,140682)

21588818884 (189848,185083)

21588818885 (191070,171269)

21588818886 (207015,67696)

21588818887 (203854,80439)

21588818888 (307884,1770)

21588818889 (209900,58099)

21588818890 (208241,63405)

21588818891 (192010,161457)

21588818892 (194387,139435)

21588818893 (205547,73293)

21588818894 (191075,171215)

21588818895 (202991,84396)

21588818896 (192289,158672)

21588818897 (232494,20007)

21588818898 (192986,151956)

21588818899 (200313,98216)

21588818900 (202944,84618)

21588818901 (237012,16565)

21588818902 (214086,46883)

21588818903 (200103,99408)

21588818904 (203511,81985)

21588818905 (194104,141864)

21588818906 (211140,54474)

21588818907 (191597,165685)

21588818908 (214213,46585)

21588818909 (197582,115129)

21588818910 (202966,84514)

21588818911 (322185,1270)

21588818912 (194740,136473)

21588818913 (317076,1425)

21588818914 (200702,96052)

21588818915 (192426,157325)

21588818916 (195063,133827)

21588818917 (190278,180079)

21588818918 (199423,103386)

21588818919 (192875,153003)

21588818920 (196264,124501)

21588818921 (212678,50343)

21588818922 (215727,43200)

21588818923 (233349,19294)

21588818924 (194183,141181)

21588818925 (192774,153963)

21588818926 (224563,28385)

21588818927 (220403,34446)

21588818928 (217691,39234)

21588818929 (194955,134705)

21588818930 (219881,35311)

21588818931 (195163,133020)

21588818932 (237971,15929)

21588818933 (200461,97386)

21588818934 (200949,94707)

21588818935 (201451,92041)

21588818936 (194377,139520)

21588818937 (190731,174977)

21588818938 (192678,154882)

21588818939 (212303,51315)

21588818940 (245582,11802)

21588818941 (193095,150936)

21588818942 (197073,118647)

21588818943 (189738,186389)

21588818944 (189550,188646)

21588818945 (190635,176044)

21588818946 (190134,181737)

21588818947 (245425,11873)

21588818948 (192078,160773)

21588818949 (214120,46803)

21588818950 (195110,133447)

...

21608597292 (202096,89251)

21608597293 (213266,49127)

21608597294 (210544,56490)

21608597295 (195707,129488)

21608597296 (195000,135134)

21608597297 (222742,31027)

21608597298 (197714,114904)

21608597299 (193636,146861)

21608597300 (194915,135832)

21608597301 (195622,130152)

21608597302 (196402,124205)

21608597303 (206332,70628)

21608597304 (201718,91178)

21608597305 (191155,171383)

21608597306 (222218,31793)

21608597307 (191257,170285)

21608597308 (199239,105100)

21608597309 (222418,31498)

21608597310 (212447,51216)

21608597311 (194032,143338)

21608597312 (203365,83123)

21608597313 (214595,45948)

21608597314 (216053,42737)

21608597315 (234319,18613)

21608597316 (197760,114593)

21608597317 (221076,33542)

21608597318 (231213,21241)

21608597319 (218728,37512)

21608597320 (203171,84027)

21608597321 (207838,65139)

21608597322 (230038,22350)

21608597323 (192755,155068)

21608597324 (333020,1010)

21608597325 (199691,102385)

21608597326 (212181,51917)

21608597327 (230616,21796)

21608597328 (193347,149496)

21608597329 (213301,49040)

21608597330 (192266,159854)

21608597331 (221682,32600)

21608597332 (231026,21413)

21608597333 (239495,15050)

21608597334 (230542,21866)

21608597335 (194521,139123)

21608597336 (211086,54925)

21608597337 (195628,130105)

21608597338 (193279,150124)

21608597339 (251796,9407)

21608597340 (202881,85400)

21608597341 (214795,45491)

21608597342 (190804,175225)

21608597343 (248669,10554)

21608597344 (193007,152667)

21608597345 (193242,150467)

21608597346 (235964,17380)

21608597347 (199666,102533)

21608597348 (228061,24377)

21608597349 (499171,101)

21608597350 (190229,181738)

21608597351 (199984,100669)

21608597352 (235914,17416)

21608597353 (223578,29850)

21608597354 (204486,78118)

21608597355 (221293,33201)

21608597356 (216472,41864)

21608597357 (197252,118083)

21608597358 (192998,152752)

21608597359 (200013,100501)

21608597360 (297398,2313)

21608597361 (208174,63984)

21608597362 (196614,122644)

21608597363 (194347,140606)

21608597364 (194583,138599)

21608597365 (199099,105958)

21608597366 (309709,1700)

21608597367 (211028,55090)

21608597368 (193434,148697)

21608597369 (267212,5549)

21608597370 (362050,580)

21608597371 (190527,178328)

21608597372 (196705,121981)

21608597373 (219270,36548)

21608597374 (193884,144643)

21608597375 (191521,167481)

21608597376 (225704,27089)

21608597377 (199057,106217)

21608597378 (201016,94888)

21608597379 (212300,51602)

21608597380 (202625,86634)

21608597381 (198417,110257)

21608597382 (249059,10402)

21608597383 (207699,65624)

21608597384 (196679,122170)

21608597385 (218412,38088)

21608597386 (192388,158644)

21608597387 (209323,60213)

21608597388 (395825,337)

21608597389 (194211,141778)

21608597390 (209893,58440)

21608597391 (191644,166193)

...

21628375733 (194738,138113)

21628375734 (208965,61698)

21628375735 (202992,85354)

21628375736 (216089,42889)

21628375737 (194756,137962)

21628375738 (195035,135646)

21628375739 (346049,781)

21628375740 (213162,49654)

21628375741 (255139,8382)

21628375742 (214868,45569)

21628375743 (247864,10927)

21628375744 (191386,169927)

21628375745 (206738,69482)

21628375746 (191987,163640)

21628375747 (208753,62393)

21628375748 (198003,113625)

21628375749 (195603,131070)

21628375750 (195340,133166)

21628375751 (203582,82591)

21628375752 (196155,126795)

21628375753 (228293,24250)

21628375754 (207358,67204)

21628375755 (192103,162458)

21628375756 (195921,128587)

21628375757 (271246,4902)

21628375758 (190526,179420)

21628375759 (192536,158133)

21628375760 (212552,51219)

21628375761 (204456,78689)

21628375762 (220723,34284)

21628375763 (214264,46968)

21628375764 (216663,41694)

21628375765 (217445,40129)

21628375766 (198420,110879)

21628375767 (244895,12231)

21628375768 (195726,130103)

21628375769 (202620,87152)

21628375770 (192800,155562)

21628375771 (189723,188846)

21628375772 (205892,72739)

21628375773 (204280,79457)

21628375774 (218032,39000)

21628375775 (220514,34625)

21628375776 (208691,62598)

21628375777 (190277,182283)

21628375778 (215694,43735)

21628375779 (190181,183401)

21628375780 (191137,172614)

21628375781 (193893,145424)

21628375782 (209580,59733)

21628375783 (192689,156637)

21628375784 (203883,81222)

21628375785 (195515,131767)

21628375786 (200024,101017)

21628375787 (199138,106331)

21628375788 (221281,33393)

21628375789 (194968,136198)

21628375790 (200965,95711)

21628375791 (212198,52154)

21628375792 (197398,117752)

21628375793 (190287,182167)

21628375794 (207975,65024)

21628375795 (192049,163007)

21628375796 (214901,45494)

21628375797 (236652,16976)

21628375798 (221066,33733)

21628375799 (201434,93189)

21628375800 (201794,91306)

21628375801 (242759,13287)

21628375802 (204227,79690)

21628375803 (203296,83917)

21628375804 (201754,91513)

21628375805 (190106,184280)

21628375806 (204800,77213)

21628375807 (199269,105525)

21628375808 (192448,159001)

21628375809 (202927,85665)

21628375810 (212929,50245)

21628375811 (209859,58866)

21628375812 (195899,128757)

21628375813 (190940,174775)

21628375814 (205840,72945)

21628375815 (200294,99460)

21628375816 (198080,113112)

21628375817 (286501,3118)

21628375818 (257686,7666)

21628375819 (286892,3084)

21628375820 (224017,29404)

21628375821 (195026,135720)

21628375822 (197542,116754)

21628375823 (193314,150695)

21628375824 (191768,165899)

21628375825 (194701,138424)

21628375826 (197442,117446)

21628375827 (197971,113839)

21628375828 (192497,158517)

21628375829 (208583,62957)

21628375830 (223309,30379)

21628375831 (238055,16029)

21628375832 (1000999,10)

The dapple of my eye

I was somewhat taken by this beautiful dappling on an apple skin. I wonder what caused it.

Tracks

Taken on my Bodie walk early Tuesday morning, these are not the sort of tracks I am used to seeing in the snow at the side of the road. The river is not that far away so perhaps a heron or egret, I will guess. The Old/Middle French pie de grue is the origin of the word pedigree and the tracks naturally reminded me of the fact.

Update: It appears Laurie Mace, who lives on this stretch of road, had captured a photo of the culprit:

⁦⁦@metromorning⁩ an attempted break in across the street from us a few weeks ago. pic.twitter.com/N2gQSk6prN

— Denison Denizen (@LaurieMace) March 21, 2022