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...

A million-digit Leyland prime (ryanp)

Ryan Propper ("ryanp" on Mersenne forum) had not contributed any Leyland primes prior to Wednesday, when he proclaimed this 1433792-digit integer to be a PRP. Then yesterday, he added another (at a mere 582101 digits) that had a small y [L(x,y) defines a Leyland integer as x^y+y^x, x≥y; here y=2]. The current top-five Leyland prime leaderboard now sports three million-digit Leyland primes (the first column is the number of digits):

1433792    (300102,59935)   Ryan Propper   May 2023

1000175    (218767,37314)   Gabor Levai    Mar 2023

1000027    (211185,54364)   Gabor Levai    May 2022

 582101   (1933695,2)       Ryan Propper   May 2023

 506429    (107890,49423)   Miklos Levai   Feb 2022

A million-digit Leyland prime (ramp-up)

This is an update to my previous "retry" post, wherein I announced a new million-digit Leyland prime search attempt. Today I finished ramping up the search from the initial 12 processes on 3 computers (covering 20% of the search space) to  57 processes on 18 computers (covering about 70% of the search space). My search last year had the same 18 machines doing 108 processes but I subsequently discovered that this overloading of processes was highly inefficient and detrimental to the effort. My current 57 processes should all be done (roughly) in early November and I can assign the remaining 30% of the search space — process by process, as they come due — at that time.

A million-digit Leyland prime (retry)

After my last disappointing attempt, I have started again today on a new search. Initially, I will commit twelve processes (on my fastest three computers) to about 20% of the search space, with a completion date of late October. I'm hoping for better luck this time around.

Bucket list #3

My original five-piece bucket list is here with #4 here.

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

1300 Weston Road

10-piece 'original' bucket; large fries, cole slaw, potato salad: $41.80, which included a $3
charge to guarantee 7 'white-meat' pieces, but I got only 6 (plus a drumstick and 3 wings)

All around my hat

It has been a long time since we were all excited by the aperiodic tilings of Penrose's kites and darts. There was even a version using images of chickens:

Penrose chickens

There is now a new tiling that reduces the number of necessary shapes from two to one, the first true aperiodic monotile:

Brad Klee: headliner

A million-digit Leyland prime (lghu)

Twenty-five days after I started my million-digit Leyland prime search last year, Gabor Levai — who goes by the handle "lghu" on Mersenne forum: small L, not capital i — discovered a 1000027-digit example that was in my purview.

He's done it again! This morning, eighteen days after I started my new run, Gabor posted a 1000175-digit, new prime that was in the range of Leyland number pairs I was prepared to look at this year. In fact, when I was sieving my pairs I had a choice of going low (1000100 digits up to 1000200 digits) or going higher (as high as 1000900 digits up to 1001000 digits) and I chose the lower range as I had convinced myself that Gabor was probably not engaged in a like-minded endeavour.

The advantage for me of going low is that my searching is exhaustive. Having already looked at all Leyland number pairs of 1000000 digits up to 1000100 digits, finding a new PRP in the range of 1000100 digits up to 1000200 digits would allow me to (eventually) say that that the new PRP and the previous 1000027-digit one are consecutive Leyland primes.

Update (March 15): I have ceased my new run on the understanding that Gabor will himself check all of the intervening Leyland candidate pairs for primality. Indications are that he can do this faster than I. Coincidentally, I found a new 386805-digit prime early this morning, the first such since 2 May 2022.

A million-digit Leyland prime (start of a new run)

Five days ago I started a new run of testing candidates for the property of being a million-digit Leyland prime. The million-digit part is relatively easy; the primality testing, not so much. My last run required nine months, not counting the month it took to sieve. I'm not pushing it for now. Of the 59364 candidates in this run, I'm only doing 9000 on my three iMacs. This should be done in July. My Mac minis are testing much smaller Leyland numbers (currently ~386750 digits) for primality and I'll keep that going until I exhaust my current crop of sieved numbers in that range. Only then will I divert them to help in the million-digit hunt.

Small-string final non-appearance coincidences in base-ten powers of two

If we look at OEIS A094776, one sees the beginnings of sequences that apply the inherent concept to strings of more than one digit. Keith Lynch suggested the idea (tongue-in-cheek, I thought) on MathFun a few days ago and Maximilian Hasler actually worked out the numbers for strings 10 to 18. I decided to chart a more comprehensive listing...

My results: sorted by strings and sorted by powers of two. There are "coincidences" where two or more strings share the same power of two as their final non-appearance exponent (for example, the two 71s in A094776 for digits 5 and 7). I'll list those here after brief summaries of each n-digit result:

1-digit strings

{71,5}
{71,7}
...
range: 119.5 ± 48.5
average: 1026/10
...
{153,3}
{168,2}

string coincidence

71: {5,7}

2-digit strings

{1300,91}
{1416,07}
...
range: 2399.5 ± 1099.5
average: 215386/100
...
{3493,28}
{3499,95}

string coincidence

2146: {33,48}

3-digit strings

{20589,141}
{20729,713}
...
range: 37290.5 ± 16701.5
average: 28860154/1000
...
{51375,552}
{53992,661}

string coincidences

22044: {024,275}
24486: {404,675}
25305: {410,947}
25440: {317,604}
25668: {442,815}
25704: {123,766}
25980: {096,868}
26046: {378,588}
26136: {422,677}
26316: {227,929}
26477: {152,690}
26695: {085,256}
26792: {048,732}
27003: {737,974}
27121: {545,932}
27479: {183,687}
28196: {300,554}
28252: {116,641}
28270: {099,575}
28317: {578,656}
28425: {287,392}
28532: {171,910}
28609: {017,919}
28784: {033,719}
28850: {164,647}
28891: {346,505}
29173: {648,787}
29705: {668,997}
29711: {335,799}
29976: {665,995}
30977: {131,395}
32637: {076,426}
33550: {555,796}
33607: {582,598}
33631: {117,735}
39571: {021,622}

4-digit strings

{266774,6355}

{273317,4949}

...

range: 459917 ± 193143

average: 3642758125/10000

...

{625874,6636}

{653060,6164}

string coincidences

296341: {0317,9730}

299582: {2041,2103}

302068: {8362,9844}

303160: {1883,2335}

304631: {5262,7194}

304639: {2896,5816}

305663: {5142,5397}

307552: {0904,8968}

307739: {9064,9656}

308220: {2396,6102}

308408: {5509,8025}

308917: {0658,5181}

309422: {2879,8852}

310452: {2800,6576}

310789: {0093,5900}

311943: {4436,7337}

312480: {5954,9044}

312956: {1741,1944}

313628: {7855,9582}

314127: {4003,6689}

315423: {6059,6252}

315844: {3119,7155}

316864: {2825,4655}

317042: {0507,0655}

318106: {7886,9390}

318428: {1872,4548}

318594: {5091,5501}

318703: {0505,6814}

319722: {7009,7541}

320081: {0089,1201}

320549: {4780,9688}

321025: {0020,8594}

321026: {1045,1488}

321108: {2364,3963}

322286: {5390,9872}

322907: {5976,9894}

323025: {3082,5477}

323178: {4438,7411,9268}

323872: {5264,8601}

323973: {7651,9051}

324814: {4462,4521}

324885: {4116,8104}

326420: {5641,8345}

326585: {0703,3413}

326793: {7640,9369}

326886: {3632,7622}

327005: {7087,8412}

327241: {0013,4113}

327387: {2757,6279}

327406: {8469,9238}

327692: {0328,1158}

327856: {6037,7447}

327899: {2569,8404}

328022: {2068,8665}

328433: {3715,9685}

328441: {4541,5491}

328592: {1770,7600}

329117: {2725,3266,7959}

329142: {1846,7235}

329285: {0324,7893}

329773: {1510,6807}

329834: {3380,5873}

330434: {5197,8702}

330442: {0232,8274}

330690: {4111,6761}

331061: {3225,5929}

331358: {2540,7253}

331463: {1493,9185}

331571: {1174,7276}

331610: {8690,9273}

332045: {1192,9701}

332136: {2114,8049}

332481: {1728,5967}

332507: {4609,5093}

332627: {0126,2700}

333146: {2636,5860}

333450: {8609,9537}

333462: {5600,8478}

333663: {1050,7693}

333682: {8270,9353}

333998: {4034,7687}

334303: {3254,4714}

334617: {6586,7828}

334837: {2030,8091}

334914: {9034,9881}

335184: {3039,6080}

335216: {5066,9474}

335580: {4004,4864}

335618: {5891,9991}

335647: {4715,9359}

335648: {1060,3150}

335833: {0604,9269}

335867: {5145,8779}

336146: {7404,8354}

336846: {2783,4059}

337394: {0750,7595}

337524: {5159,9199}

338108: {4509,6866}

338157: {2741,4933}

338377: {1494,2094}

338420: {1515,3966}

338794: {9594,9845}

339154: {0743,9414}

339356: {1476,5817}

340009: {2933,8133}

340042: {2953,9957}

340097: {0172,1204}

340281: {5404,8419}

340472: {0060,3472}

340926: {0872,6788}

341131: {0237,6434,8154}

341157: {1810,9578}

341370: {5676,5786}

341506: {1432,6014}

341528: {3645,7689}

341902: {2203,6407}

341914: {5194,8521}

341983: {6118,7811}

342019: {2264,5218}

342233: {3949,5461}

342260: {0897,8605}

342394: {0260,8494}

342564: {4002,9173}

342630: {5527,8773}

342757: {3569,9032}

342821: {4799,5993}

343798: {7343,8491}

344003: {5961,7381}

344041: {2518,3260}

344251: {0102,3745}

344396: {1523,4810}

344780: {9627,9882}

345245: {1558,7972}

345278: {8163,9325}

345602: {4701,5555}

345933: {4788,9070}

345936: {1277,4166}

346134: {2839,8391}

346151: {0856,3654}

346219: {5495,5683}

346364: {4192,7639}

346769: {0587,1635,2374}

347036: {0559,8406}

347161: {8907,9741}

347522: {3605,3996}

347620: {2511,6244}

347740: {0579,8606}

347746: {4358,8878}

347775: {8741,9906}

347907: {2204,9640}

347925: {5423,6137}

348332: {6774,8667}

348762: {3488,6962}

348855: {0956,2340}

348911: {3646,9581}

349048: {3546,3744}

349073: {4179,8338}

349254: {3805,6726}

349471: {1047,4986}

349829: {6157,7860}

350183: {2142,6961,9708}

350308: {4130,6670}

350516: {4296,8527}

350569: {4060,9635}

350635: {1378,2927}

350684: {1261,7773}

351243: {1769,3536}

351326: {3121,7915}

351529: {2380,6571}

351561: {3911,5291}

351713: {2415,8585}

352134: {0692,1374}

352235: {5647,6629}

352294: {7046,8069}

352417: {0619,5419}

352438: {2439,8199}

352539: {6748,8705}

352694: {1180,7163}

353011: {2131,6621}

353082: {0174,4600}

353168: {1889,5682}

353239: {4050,9956}

353464: {2294,4966}

353828: {4931,7372}

353840: {0217,0907}

354455: {3776,6185}

355087: {4417,6577}

355188: {3510,5193}

355324: {0017,2318}

355402: {6057,7503}

355434: {3307,6790}

355805: {2138,3581}

355947: {1746,6743}

356094: {2159,6092}

356103: {2035,2615}

356255: {7001,8420}

356497: {1586,2339}

356516: {0066,0762}

356620: {3657,4174}

356900: {2055,5223}

356973: {8776,9103}

357223: {1687,7664}

358009: {7382,7497}

358487: {1366,4133}

359062: {1146,8147}

359220: {3660,8822}

359230: {6165,8191}

359616: {0633,6176}

360109: {2307,7362}

360826: {3455,7078}

361585: {4828,9217}

361678: {3837,7685}

362060: {0732,5850}

362094: {4434,8748}

362106: {2659,8340}

362129: {1302,2693,3550,9945}

362530: {2483,3015}

362580: {1281,5708}

362896: {4255,7722}

362924: {4744,9644}

363104: {0674,3672}

363222: {3484,7142}

363547: {2239,5260}

363885: {6291,7596}

363897: {2848,4773}

364027: {1035,8633}

364036: {7200,8926}

364503: {0589,5054}

364986: {0798,8765}

365010: {0005,8973}

365249: {5474,7459}

365443: {0073,7884}

365632: {1824,7237}

365776: {0071,8921}

365966: {0244,7492}

365968: {4984,5128}

365996: {7230,9404}

366225: {1300,3345}

366355: {9461,9915}

367072: {1481,5969}

367111: {2634,6306}

367217: {2393,2429}

367898: {4498,8180}

368443: {5085,7256}

368694: {5507,8283,9984}

368976: {6996,7062,7667}

369313: {2716,8640}

369495: {4545,7392}

369970: {2764,5011}

370814: {4601,6747}

370974: {4575,5613}

371553: {2804,6497}

372064: {6702,8361}

372108: {2949,7989}

372252: {1248,7907}

372311: {0050,5804}

372356: {4343,8399}

372357: {1716,7165}

372814: {6287,7363}

373523: {4530,8105}

373861: {3379,4697}

374797: {5925,8296}

374810: {1433,9562}

375824: {2284,6196}

376003: {1052,8083}

376138: {4294,7424}

376739: {9078,9973}

376760: {2051,9750}

376785: {0438,7605}

377455: {4277,5315}

378037: {2196,7421}

378355: {2938,9245}

378973: {2147,9852}

378984: {0276,6941}

379462: {0014,8562}

379612: {0680,5393}

379782: {1791,2266}

380544: {1328,8863}

381077: {7870,9296}

381177: {2894,7066}

381669: {0006,5820}

381849: {0597,5625}

382182: {3690,4675}

382940: {3346,8450}

383096: {4242,6254}

383242: {2584,7346}

383783: {3804,9176}

383870: {3557,9628}

384280: {1068,7269}

384385: {2436,4817}

384511: {3194,6645}

384649: {0881,6310}

385168: {4726,5116}

385543: {2545,8762}

386204: {9665,9773}

386294: {0727,1169}

387984: {5505,9448}

388202: {5985,8507}

389330: {1857,9447}

389349: {2841,5269}

389481: {2729,3998}

389795: {2367,4654}

389853: {1809,9849}

390037: {5162,5844}

390708: {0628,9706}

391278: {5899,7553}

391837: {2013,8526}

391929: {1713,3793}

392002: {2034,2872}

392518: {7118,9658}

393878: {5496,9517}

396142: {6936,8249}

396567: {7081,8211}

396886: {3938,8909}

397399: {0352,2680}

397503: {5059,9082}

397631: {1691,7111}

398347: {1367,5590}

398451: {2077,3116}

398583: {0687,9055}

398635: {4387,6361}

401346: {0266,4432}

402139: {5528,6893}

402315: {0936,7464}

403677: {3809,8263}

405493: {6027,6698}

407926: {0891,6640}

412908: {0278,3055}

413260: {0749,9602}

413732: {6654,7788}

414179: {8067,8580}

415732: {0870,6078}

416017: {0320,5720}

417139: {1126,9616}

417664: {4785,6195}

419623: {3402,3883}

420795: {1676,6722}

421861: {0786,1291}

423699: {3309,8888}

426750: {7638,9766}

433287: {1151,7375}

441637: {2651,2822}

452674: {3564,8663}

504659: {3418,3774}

514562: {0052,1787}

The red numbers in the summaries for each n-digit result has become a sequence: A360623.

Fenestron

The loud sound of a helicopter just before 10:30 p.m. last night got me to look out the window and, because of the unusual appearance of the tail, run to get my camera and take a photo from the front porch. The aircraft disappeared behind the apartment buildings on Weston Rd. — seemingly landing (in the vicinity of Weston's UP Express train station; I'm going to guess at the Toronto Paramedic Services lot). Coincidentally, a UP Express service alert appeared one-and-a-half hours later.