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.

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.

Lake effect

click to enlarge

It is sunny against a mostly blue sky backdrop outside but, as predicted, very cold. I did not even take Bodie for his walk this morning! My GOES 16 screen grab (above) shows snow streamers coming off the Great Lakes (left) with the prevailing northwesterly winds (below). The situation is even more pronounced off the coast of Maine (far right) where the lake effect is an ocean effect.

click to enlarge

Odd integer multiples via chess Knight moves

three: click to enlarge

five: click to enlarge

seven: click to enlarge

A million-digit Leyland prime (end of run)

manual distribution worksheet for 59536 primality tests

An hour ago I completed the primality check of 59536 Leyland-prime candidates that I started on 2 May 2022. My 27 May 2022 reality check explains why it took way longer (just short of nine months) than I had originally planned. Preceding the primality testing was another month (or so) sieving the original Leyland numbers file (so as to exclude divisibility by primes up to 2*10^11), so ten months altogether, now done.

What have I discovered? Of Leyland numbers with at least one million decimal digits, but fewer than one million one hundred decimal digits, there is only one prime. That prime was discovered by Gabor Levai long before I got to it. I saved all of my primality-test output where the 59536 entries are listed smallest to largest. If you want to see the one prime, search for "PRP".

The execution times vary wildly (due to processor circumstances) with an average of 9.45 hours per test. That would work out to 64 years if I hadn't been able to multi-process. I know now how to keep the execution times to 6 hours or less per test but that means running fewer processes per machine. Still, I might be able to shave a couple of months off the total time required for the next run.

A digit-spine sequence

Éric Angelini presented a proposal for three "digit-spine" sequences on his blog here, as well as to the MathFun community. I decided to take on the first one:

s = 1, 10, 2, 0, 3, 26, 9, 119, 532, 4, 6, 896, 118, 34, 15, ...

p = 2, 11, 2, 2, 3, 23, 7, 113, 523, 3, 5, 887, 113, 31, 13, ...

d = 1,  1, 0, 2, 0,  3, 2,   6,   9, 1, 1,   9,   5,  3,  2, ...

Our sought-after sequence is 's'. This is followed by 'p', a sequence of examples (for each, there might be a second such) of primes that are as close to the terms of 's' as possible. Finally 'd', the absolute differences between the respective  's' and 'p' values. Determining 's' is that it must be the lexicographically earliest sequence of distinct nonnegative integers such that its digit-spine (the sequential digits) as well as the digit-spine of 'd' are identical.

For example, the third 'd' being 0 is a consequence of the zero in 10, the second term of 's'. This forces the third term of 's' to be the smallest prime (as it had not already been used). And so on. There will always be integers in 'd' that inform our upcoming values in 's'. And the digits of those values get added to the growing end of our 'd' sequence.

So far, all of our 'd' values are single digits. But there will come a point when concatenating a digit with the next one in line allows for a smaller 's' term than what is available for the single digit. This first happens at term #1622 where a d = 1 dictates s = 1010. The next d digit is 0. Concatenating that 0 with the previous 1 gives us d = 10 at #1622 and this allows the smaller s = 897. This makes me want to know at which term the first three-digit d appears.

Maximilian Hasler helped Éric get the sequence into the OEIS. I have an indexed listing up to one million terms here.

10*987/6+54/3*21

Happy New Year!