Tuesday, August 27
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):
Saturday, August 24
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.
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
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
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
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!
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
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
2 L(34684,105) - L(29356,257) 2887602 <70425> 12 Oct 25
3 L(29356,257) - L(30280,241) 6274269 <71439> 26 Nov 20
4 L(30280,241) - L(104824,5) 5256668 <72700> 23 Dec 13
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. It's best for now to think of the completion dates as the earliest possible completion dates.
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 1
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.
Tuesday, July 9
Irvin K. Wilhelm and why Wikipedia sucks
The first thing to note is that when you search for Irvin K. Wilhelm in Wikipedia, it redirects to Kaiser Wilhelm (baseball). They had to add the word 'baseball' to distinguish the man from the real Kaiser Wilhelm to which the then-applied name referred: German emperor Wilhelm II. Make no mistake, 'Kaiser' during World War I was a derogatory moniker and calling Irvin this name was not meant to endear him to you but, rather, to get under his skin. Irvin abhorred it! It strikes me as unconscionable that (83 years after Irvin's death) Wikipedia continues to perpetuate the pejorative as their gateway for the man. I guess (a century after World War I) people no longer relate to the vitriol once hurled. I tried on 8 July 2011 to change this sad state of affairs but a Spanneraol reverted my edit (thirty minutes after I made it) because Kaiser Wilhelm "is the most common name used for him, MLB and baseball reference use it and those are the sources we go by." Wow! Imagine if the gateway entry for Mary Mallon was Typhoid Mary because — well — who the heck knows Mary Mallon?
The second thing to note is that Wikipedia's date of birth for Irvin has been 26 January 1874 from the article's inception on 26 July 2005 to now, with a brief respite to reality from 21 August 2008 (when I changed the date to 1877, Irvin's actual birth year) until 7 January 2009 (when a Woohookitty changed it back to 1874 as "per several sources, namely the baseball cube and baseball-reference.com"). I discovered the unfortunate reversion on 8 July 2011, on which date I attempted again to correct the record. I noted: "Any baseball site still listing his birth year as 1874 is perpetuating an error." This time I added a reference to the 1900 census clearly indicating his birth year as 1877 and his age as 23. The edit lasted less than two hours. Spanneraol (yes, the same person that thinks it's OK to perpetuate a personal invective because it is common) reverted the edit. No reason was given.
There is no question that Irvin Key Wilhelm was born on 26 January 1877. In addition to the 1900 census noted above, there is (most importantly) his Wayne County birth record. In the 1880 census he is, unsurprisingly, three. Irvin's 1902 marriage record has his age as 23, which would have had him born in 1879. The 1910 census would have had him born in 1878. His 1918 draft registration lists his birth year as 1879. The 1930 census would have had him born in 1879. So the evidence suggests that around 1901, Irvin managed to get a couple of years younger! According to Gregory H. Wolf, this age reduction was actually a longstanding tradition among ballplayers. Wolf's Wilhelm-biography for the Society for American Baseball Research is excellent — even if the header is 'Kaiser Wilhelm'. He uses 1877 for the birth year but footnotes: "baseball-reference.com gives the date 1874". I think that this is no longer the case.
Irvin K. Wilhelm died at the age of 59. Wikipedia would have his age have been 62. As bad as that is, I'll point out that Wilhelm's short obit in the New York Times stated that he was 53 (and misspelled his name Irving to boot). Ouch!
Friday, May 24
Prime sandwiches (made with one-derbread)
This refers to David James Sycamore's A306861, which asks — for each prime — how many digit 1s must be appended to both sides of that prime (even if it already starts or ends with one or more 1s) before we generate another prime. For example, for the prime 3 only one 1 is required: 3 → 131. For the prime 263, ten are required: 263 → 11111111112631111111111. For the prime 34337, it takes one hundred 1s on each side, and for the prime 36161, it takes one thousand. Here's a three-column list giving the number for all primes less than 10000:
2 * 2833 5 6311 6
3 1 2837 3 6317 1
5 1 2843 1 6323 10
7 3 2851 3 6329 19
11 * 2857 2 6337 38
13 3 2861 6 6343 935
17 1 2879 3 6353 3
19 21 2887 1368 6359 4
23 1 2897 1 6361 3
29 1 2903 18 6367 44
31 2 2909 3 6373 2915
37 * 2917 12 6379 5
41 3 2927 4 6389 33771
43 2 2939 7 6397 2
47 1 2953 2 6421 47850
53 1 2957 18 6427 50
59 42 2963 1 6449 10
61 14 2969 7 6451 2
67 3 2971 11 6469 8
71 73 2999 1914 6473 >50000
73 3 3001 3 6481 78
79 2 3011 4 6491 1
83 1 3019 3 6521 1
89 4 3023 4 6529 2
97 3 3037 5 6547 2
101 * 3041 1 6551 1
103 2 3049 591 6553 2
107 1 3061 6 6563 18
109 3 3067 6 6569 10
113 1 3079 2 6571 3
127 3 3083 28 6577 18
131 1 3089 3 6581 1
137 3 3109 2 6599 3
139 3 3119 12 6607 8
149 1 3121 2 6619 2
151 6 3137 1 6637 6
157 2 3163 11 6653 58
163 3 3167 1 6659 27
167 192 3169 2 6661 2
173 1 3181 5 6673 27
179 4 3187 2 6679 2
181 3 3191 288 6689 3
191 3 3203 3 6691 2
193 8 3209 7 6701 73362
197 1 3217 96 6703 81
199 9 3221 7 6709 18
211 36 3229 159 6719 1
223 5 3251 1 6733 6
227 12 3253 9 6737 7
229 5 3257 3 6761 1
233 18 3259 2 6763 12
239 1 3271 5 6779 54
241 26 3299 16 6781 2
251 1 3301 5 6791 1
257 16 3307 102 6793 593
263 10 3313 2 6803 9
269 15 3319 3 6823 2
271 2 3323 3 6827 52
277 72 3329 10 6829 3
281 22 3331 2 6833 1
283 3 3343 20 6841 2
293 4 3347 18 6857 9
307 2 3359 4 6863 1
311 4 3361 5 6869 4
313 5 3371 1 6871 2
317 1 3373 141 6883 2
331 12 3389 15 6899 1
337 5 3391 5 6907 26
347 13 3407 21 6911 1
349 3 3413 7 6917 3
353 9 3433 12 6947 1
359 1 3449 12 6949 20
367 6 3457 5 6959 1
373 60 3461 13 6961 6
379 2 3463 2 6967 11
383 1 3467 3 6971 39
389 58 3469 39 6977 1413
397 >75000 3491 36 6983 1
401 1 3499 18 6991 44
409 2 3511 8 6997 2
419 24 3517 >50000 7001 3
421 42 3527 1 7013 30
431 4 3529 5 7019 984
433 6 3533 3 7027 3
439 30 3539 1 7039 2
443 1 3541 32 7043 15
449 4 3547 5 7057 2
457 3 3557 1 7069 12
461 27 3559 5 7079 7
463 3 3571 2 7103 54
467 67 3581 15 7109 1
479 3 3583 2 7121 3
487 5 3593 3 7127 1
491 12 3607 6 7129 35
499 6 3613 42 7151 3
503 1 3617 6 7159 2
509 1 3623 13 7177 3
521 12 3631 3 7187 696
523 6 3637 16214 7193 6
541 6 3643 437 7207 6
547 11 3659 255 7211 6
557 145 3671 1 7213 582
563 >60000 3673 5 7219 2
569 3 3677 6 7229 3
571 2 3691 17 7237 6
577 2 3697 3 7243 5
587 9 3701 33 7247 78
593 187 3709 6 7253 6
599 1 3719 1 7283 10
601 2 3727 7211 7297 6
607 38 3733 5 7307 31
613 117 3739 3 7309 20
617 6 3761 3 7321 3
619 3 3767 21 7331 24
631 2 3769 5 7333 2
641 1 3779 1 7349 1
643 5 3793 8 7351 5
647 3 3797 42 7369 8
653 73 3803 3 7393 2
659 312 3821 12 7411 3
661 3 3823 150 7417 2
673 3746 3833 3 7433 1
677 25 3847 18 7451 4
683 1 3851 1 7457 1
691 5 3853 1103 7459 11
701 1 3863 7 7477 2
709 45 3877 204 7481 74703
719 1 3881 13 7487 6
727 572 3889 3 7489 3
733 42 3907 3 7499 1
739 46026 3911 552 7507 24
743 1 3917 3 7517 3
751 3 3919 20 7523 132
757 2 3923 6 7529 1
761 42 3929 1 7537 255
769 2 3931 14 7541 1
773 6 3943 3 7547 7
787 2 3947 3 7549 5
797 1 3967 11 7559 12
809 16 3989 1 7561 3
811 5 4001 3 7573 3
821 1 4003 5 7577 16
823 3 4007 1 7583 6
827 10 4013 4 7589 1
829 12 4019 1 7591 5
839 15 4021 5 7603 3
853 144 4027 6 7607 9
857 2068 4049 48 7621 15
859 5 4051 6 7639 3
863 9 4057 29 7643 1
877 18 4073 1 7649 3
881 3 4079 27 7669 23
883 8 4091 6 7673 13
887 4 4093 336 7681 3
907 3 4099 2 7687 3
911 4 4111 9 7691 3
919 5 4127 90 7699 60
929 12 4129 2 7703 4
937 2 4133 4 7717 18
941 16 4139 9 7723 1293
947 1 4153 8 7727 4
953 1 4157 21 7741 5
967 6 4159 >50000 7753 2
971 4 4177 57 7757 7
977 166 4201 2 7759 5
983 3 4211 1 7789 2
991 24 4217 22 7793 3
997 2 4219 3 7817 3
1009 447 4229 16 7823 1
1013 138 4231 804 7829 6
1019 4 4241 12 7841 10
1021 23 4243 3 7853 1
1031 1 4253 7 7867 15
1033 2 4259 1 7873 14
1039 11 4261 42 7877 12
1049 1 4271 1 7879 12
1051 3 4273 5 7883 1
1061 4 4283 3 7901 9
1063 12 4289 72 7907 13
1069 3 4297 2 7919 21
1087 6 4327 3 7927 810
1091 6 4337 3 7933 2
1093 14 4339 3 7937 7
1097 141 4349 3 7949 3
1103 1 4357 125 7951 3
1109 1 4363 2 7963 2
1117 41 4373 6 7993 2
1123 11 4391 42 8009 3
1129 8 4397 1 8011 11
1151 133 4409 49 8017 3
1153 3 4421 16 8039 1
1163 804 4423 2 8053 6
1171 173 4441 53 8059 126
1181 6 4447 26 8069 3
1187 1 4451 1 8081 1
1193 4 4457 591 8087 1
1201 5 4463 10 8089 492
1213 5 4481 12 8093 6
1217 3 4483 317 8101 5
1223 9 4493 1 8111 10398
1229 1 4507 30 8117 9
1231 3 4513 2 8123 4
1237 9 4517 9 8147 13
1249 * 4519 2 8161 2
1259 27 4523 4 8167 9
1277 1 4547 1 8171 1
1279 3 4549 6 8179 90
1283 1 4561 2 8191 5
1289 3 4567 323 8209 3
1291 173 4583 303 8219 3
1297 3 4591 2 8221 6575
1301 1 4597 2 8231 24
1303 8 4603 6 8233 3
1307 780 4621 27 8237 7953
1319 432 4637 30 8243 1
1321 6 4639 6 8263 150
1327 15 4643 3 8269 5
1361 4 4649 33 8273 216
1367 18 4651 5 8287 29
1373 1 4657 710 8291 6
1381 12 4663 23 8293 11
1399 17 4673 3 8297 4
1409 4 4679 4 8311 2
1423 2 4691 45 8317 20
1427 4 4703 1 8329 3
1429 12 4721 1 8353 3
1433 15 4723 2 8363 24
1439 4 4729 3 8369 1
1447 8 4733 1 8377 188
1451 12 4751 3 8387 1
1453 6 4759 2 8389 5
1459 17 4783 17 8419 3
1471 3 4787 51 8423 1
1481 252 4789 29 8429 1
1483 3 4793 30 8431 6
1487 72 4799 7 8443 12
1489 5 4801 54 8447 3
1493 16 4813 2 8461 3
1499 121 4817 1 8467 18
1511 36 4831 20 8501 4
1523 3 4861 563 8513 1
1531 6 4871 1 8521 6
1543 5 4877 2988 8527 6
1549 5 4889 1 8537 1
1553 3 4903 23 8539 3
1559 6 4909 47 8543 6
1567 5 4919 3 8563 2
1571 48 4931 6 8573 9
1579 12 4933 2 8581 3
1583 1 4937 1 8597 1
1597 15 4943 12 8599 18
1601 3 4951 3 8609 18
1607 9 4957 12 8623 342
1609 3 4967 16 8627 1
1613 1 4969 3 8629 2
1619 1 4973 1 8641 54
1621 17 4987 17 8647 3
1627 54 4993 3 8663 7
1637 1 4999 2 8669 13
1657 5 5003 6 8677 2
1663 5 5009 1 8681 6
1667 25 5011 2 8689 2
1669 11 5021 1 8693 9
1693 2 5023 2 8699 1554
1697 46 5039 7 8707 5
1699 2 5051 3 8713 5
1709 3 5059 60 8719 3
1721 7 5077 120 8731 2
1723 2 5081 7 8737 5
1733 1 5087 537 8741 570
1741 6 5099 1 8747 1
1747 8 5101 11 8753 1
1753 113 5107 2 8761 48
1759 2 5113 48 8779 2
1777 2 5119 3 8783 3
1783 12 5147 1 8803 2
1787 36 5153 1 8807 18
1789 14 5167 2 8819 9
1801 2 5171 483 8821 3
1811 3078 5179 2 8831 1
1823 9565 5189 4 8837 72
1831 3 5197 42 8839 5
1847 1 5209 2 8849 1
1861 2 5227 2 8861 4
1867 2 5231 1 8863 5
1871 36 5233 5 8867 3
1873 6 5237 4 8887 3
1877 6 5261 18 8893 30
1879 6 5273 3 8923 8
1889 1 5279 1 8929 33
1901 3 5281 71 8933 3
1907 4 5297 15 8941 3
1913 1 5303 16 8951 6
1931 1 5309 10 8963 702
1933 3 5323 378 8969 1
1949 34 5333 18 8971 18
1951 8 5347 6 8999 7
1973 138 5351 1 9001 57
1979 6 5381 186 9007 2
1987 3 5387 1 9011 6
1993 6 5393 7 9013 11
1997 1 5399 1 9029 60
1999 479 5407 42 9041 30
2003 40 5413 2 9043 3
2011 2 5417 6 9049 2
2017 2 5419 50 9059 1
2027 10 5431 5 9067 261
2029 311 5437 3 9091 17
2039 1 5441 4 9103 5
2053 83 5443 47 9109 24
2063 228 5449 12 9127 3
2069 1 5471 3 9133 3
2081 1 5477 426 9137 3
2083 2 5479 14 9151 72
2087 1 5483 3 9157 2
2089 6 5501 42 9161 3
2099 3 5503 107 9173 226
2111 12 5507 18 9181 8
2113 8 5519 1 9187 6
2129 1 5521 3 9199 78
2131 5 5527 5 9203 42
2137 2 5531 639 9209 1
2141 18 5557 359 9221 4
2143 2 5563 30 9227 1
2153 1 5569 2 9239 1
2161 6 5573 1 9241 2
2179 2 5581 3 9257 1
2203 2 5591 42 9277 2
2207 63 5623 2 9281 1
2213 1 5639 7 9283 156
2221 3 5641 5 9293 1
2237 4 5647 833 9311 6
2239 5 5651 1 9319 5
2243 130 5653 6 9323 43
2251 5 5657 75 9337 20
2267 7 5659 6 9341 31
2269 2 5669 1 9343 2
2273 3 5683 3 9349 3
2281 3 5689 3 9371 36
2287 120 5693 6 9377 1
2293 3 5701 5 9391 2
2297 1 5711 6 9397 17
2309 1 5717 6 9403 2
2311 2 5737 3 9413 3
2333 6078 5741 1 9419 24
2339 13 5743 5 9421 3
2341 11 5749 3 9431 6
2347 2 5779 20 9433 8
2351 3 5783 1 9437 1
2357 3 5791 2 9439 6
2371 11 5801 24 9461 19
2377 3 5807 1 9463 18
2381 192 5813 364 9467 1
2383 6 5821 5 9473 441
2389 14 5827 5 9479 9
2393 1 5839 2 9491 1
2399 6 5843 3 9497 15
2411 198 5849 24 9511 9
2417 1 5851 8 9521 4
2423 1 5857 11 9533 3
2437 3 5861 1 9539 3
2441 3 5867 31 9547 6
2447 1 5869 47 9551 1
2459 4 5879 1 9587 3
2467 6 5881 6 9601 2
2473 5 5897 19 9613 3
2477 1 5903 55 9619 672
2503 8 5923 2 9623 15
2521 5 5927 52836 9629 1
2531 1 5939 24 9631 2
2539 5 5953 35 9643 3
2543 4 5981 1 9649 11
2549 28 5987 1 9661 2
2551 18 6007 2 9677 1
2557 2 6011 9 9679 2
2579 1 6029 3 9689 24
2591 >50000 6037 17 9697 30
2593 1332 6043 5 9719 25
2609 3 6047 43 9721 2
2617 18 6053 21 9733 66
2621 1 6067 6 9739 2
2633 48 6073 12 9743 63
2647 3 6079 17 9749 36
2657 6 6089 42 9767 9
2659 2 6091 11 9769 2
2663 1 6101 4 9781 2
2671 11 6113 6 9787 5
2677 420 6121 5 9791 228
2683 9 6131 39 9803 1
2687 6 6133 5 9811 15
2689 8 6143 3 9817 5
2693 18 6151 3 9829 2
2699 3 6163 6 9833 3
2707 30 6173 1 9839 1
2711 93 6197 1 9851 3
2713 12 6199 12 9857 1
2719 12 6203 3 9859 24
2729 1 6211 74 9871 56
2731 2 6217 45 9883 18
2741 9 6221 15 9887 7
2749 5 6229 2 9901 59
2753 6 6247 5 9907 3
2767 3 6257 16 9923 4
2777 9 6263 6 9929 24
2789 16 6269 1 9931 102
2791 3 6271 3 9941 1
2797 8 6277 2687 9949 3
2801 21 6287 190 9967 2
2803 18 6299 28 9973 2
2819 6 6301 3 etc
The five asterisks indicate infinity. No finite number of 1s will ever make those primes prime again. The six greater-than symbols show lower limits for unknowns. The largest known numbers are:
74703 @ 7481
73362 @ 6701
52836 @ 5927
47850 @ 6421
46026 @ 739
2 * 2833 5 6311 6
3 1 2837 3 6317 1
5 1 2843 1 6323 10
7 3 2851 3 6329 19
11 * 2857 2 6337 38
13 3 2861 6 6343 935
17 1 2879 3 6353 3
19 21 2887 1368 6359 4
23 1 2897 1 6361 3
29 1 2903 18 6367 44
31 2 2909 3 6373 2915
37 * 2917 12 6379 5
41 3 2927 4 6389 33771
43 2 2939 7 6397 2
47 1 2953 2 6421 47850
53 1 2957 18 6427 50
59 42 2963 1 6449 10
61 14 2969 7 6451 2
67 3 2971 11 6469 8
71 73 2999 1914 6473 >50000
73 3 3001 3 6481 78
79 2 3011 4 6491 1
83 1 3019 3 6521 1
89 4 3023 4 6529 2
97 3 3037 5 6547 2
101 * 3041 1 6551 1
103 2 3049 591 6553 2
107 1 3061 6 6563 18
109 3 3067 6 6569 10
113 1 3079 2 6571 3
127 3 3083 28 6577 18
131 1 3089 3 6581 1
137 3 3109 2 6599 3
139 3 3119 12 6607 8
149 1 3121 2 6619 2
151 6 3137 1 6637 6
157 2 3163 11 6653 58
163 3 3167 1 6659 27
167 192 3169 2 6661 2
173 1 3181 5 6673 27
179 4 3187 2 6679 2
181 3 3191 288 6689 3
191 3 3203 3 6691 2
193 8 3209 7 6701 73362
197 1 3217 96 6703 81
199 9 3221 7 6709 18
211 36 3229 159 6719 1
223 5 3251 1 6733 6
227 12 3253 9 6737 7
229 5 3257 3 6761 1
233 18 3259 2 6763 12
239 1 3271 5 6779 54
241 26 3299 16 6781 2
251 1 3301 5 6791 1
257 16 3307 102 6793 593
263 10 3313 2 6803 9
269 15 3319 3 6823 2
271 2 3323 3 6827 52
277 72 3329 10 6829 3
281 22 3331 2 6833 1
283 3 3343 20 6841 2
293 4 3347 18 6857 9
307 2 3359 4 6863 1
311 4 3361 5 6869 4
313 5 3371 1 6871 2
317 1 3373 141 6883 2
331 12 3389 15 6899 1
337 5 3391 5 6907 26
347 13 3407 21 6911 1
349 3 3413 7 6917 3
353 9 3433 12 6947 1
359 1 3449 12 6949 20
367 6 3457 5 6959 1
373 60 3461 13 6961 6
379 2 3463 2 6967 11
383 1 3467 3 6971 39
389 58 3469 39 6977 1413
397 >75000 3491 36 6983 1
401 1 3499 18 6991 44
409 2 3511 8 6997 2
419 24 3517 >50000 7001 3
421 42 3527 1 7013 30
431 4 3529 5 7019 984
433 6 3533 3 7027 3
439 30 3539 1 7039 2
443 1 3541 32 7043 15
449 4 3547 5 7057 2
457 3 3557 1 7069 12
461 27 3559 5 7079 7
463 3 3571 2 7103 54
467 67 3581 15 7109 1
479 3 3583 2 7121 3
487 5 3593 3 7127 1
491 12 3607 6 7129 35
499 6 3613 42 7151 3
503 1 3617 6 7159 2
509 1 3623 13 7177 3
521 12 3631 3 7187 696
523 6 3637 16214 7193 6
541 6 3643 437 7207 6
547 11 3659 255 7211 6
557 145 3671 1 7213 582
563 >60000 3673 5 7219 2
569 3 3677 6 7229 3
571 2 3691 17 7237 6
577 2 3697 3 7243 5
587 9 3701 33 7247 78
593 187 3709 6 7253 6
599 1 3719 1 7283 10
601 2 3727 7211 7297 6
607 38 3733 5 7307 31
613 117 3739 3 7309 20
617 6 3761 3 7321 3
619 3 3767 21 7331 24
631 2 3769 5 7333 2
641 1 3779 1 7349 1
643 5 3793 8 7351 5
647 3 3797 42 7369 8
653 73 3803 3 7393 2
659 312 3821 12 7411 3
661 3 3823 150 7417 2
673 3746 3833 3 7433 1
677 25 3847 18 7451 4
683 1 3851 1 7457 1
691 5 3853 1103 7459 11
701 1 3863 7 7477 2
709 45 3877 204 7481 74703
719 1 3881 13 7487 6
727 572 3889 3 7489 3
733 42 3907 3 7499 1
739 46026 3911 552 7507 24
743 1 3917 3 7517 3
751 3 3919 20 7523 132
757 2 3923 6 7529 1
761 42 3929 1 7537 255
769 2 3931 14 7541 1
773 6 3943 3 7547 7
787 2 3947 3 7549 5
797 1 3967 11 7559 12
809 16 3989 1 7561 3
811 5 4001 3 7573 3
821 1 4003 5 7577 16
823 3 4007 1 7583 6
827 10 4013 4 7589 1
829 12 4019 1 7591 5
839 15 4021 5 7603 3
853 144 4027 6 7607 9
857 2068 4049 48 7621 15
859 5 4051 6 7639 3
863 9 4057 29 7643 1
877 18 4073 1 7649 3
881 3 4079 27 7669 23
883 8 4091 6 7673 13
887 4 4093 336 7681 3
907 3 4099 2 7687 3
911 4 4111 9 7691 3
919 5 4127 90 7699 60
929 12 4129 2 7703 4
937 2 4133 4 7717 18
941 16 4139 9 7723 1293
947 1 4153 8 7727 4
953 1 4157 21 7741 5
967 6 4159 >50000 7753 2
971 4 4177 57 7757 7
977 166 4201 2 7759 5
983 3 4211 1 7789 2
991 24 4217 22 7793 3
997 2 4219 3 7817 3
1009 447 4229 16 7823 1
1013 138 4231 804 7829 6
1019 4 4241 12 7841 10
1021 23 4243 3 7853 1
1031 1 4253 7 7867 15
1033 2 4259 1 7873 14
1039 11 4261 42 7877 12
1049 1 4271 1 7879 12
1051 3 4273 5 7883 1
1061 4 4283 3 7901 9
1063 12 4289 72 7907 13
1069 3 4297 2 7919 21
1087 6 4327 3 7927 810
1091 6 4337 3 7933 2
1093 14 4339 3 7937 7
1097 141 4349 3 7949 3
1103 1 4357 125 7951 3
1109 1 4363 2 7963 2
1117 41 4373 6 7993 2
1123 11 4391 42 8009 3
1129 8 4397 1 8011 11
1151 133 4409 49 8017 3
1153 3 4421 16 8039 1
1163 804 4423 2 8053 6
1171 173 4441 53 8059 126
1181 6 4447 26 8069 3
1187 1 4451 1 8081 1
1193 4 4457 591 8087 1
1201 5 4463 10 8089 492
1213 5 4481 12 8093 6
1217 3 4483 317 8101 5
1223 9 4493 1 8111 10398
1229 1 4507 30 8117 9
1231 3 4513 2 8123 4
1237 9 4517 9 8147 13
1249 * 4519 2 8161 2
1259 27 4523 4 8167 9
1277 1 4547 1 8171 1
1279 3 4549 6 8179 90
1283 1 4561 2 8191 5
1289 3 4567 323 8209 3
1291 173 4583 303 8219 3
1297 3 4591 2 8221 6575
1301 1 4597 2 8231 24
1303 8 4603 6 8233 3
1307 780 4621 27 8237 7953
1319 432 4637 30 8243 1
1321 6 4639 6 8263 150
1327 15 4643 3 8269 5
1361 4 4649 33 8273 216
1367 18 4651 5 8287 29
1373 1 4657 710 8291 6
1381 12 4663 23 8293 11
1399 17 4673 3 8297 4
1409 4 4679 4 8311 2
1423 2 4691 45 8317 20
1427 4 4703 1 8329 3
1429 12 4721 1 8353 3
1433 15 4723 2 8363 24
1439 4 4729 3 8369 1
1447 8 4733 1 8377 188
1451 12 4751 3 8387 1
1453 6 4759 2 8389 5
1459 17 4783 17 8419 3
1471 3 4787 51 8423 1
1481 252 4789 29 8429 1
1483 3 4793 30 8431 6
1487 72 4799 7 8443 12
1489 5 4801 54 8447 3
1493 16 4813 2 8461 3
1499 121 4817 1 8467 18
1511 36 4831 20 8501 4
1523 3 4861 563 8513 1
1531 6 4871 1 8521 6
1543 5 4877 2988 8527 6
1549 5 4889 1 8537 1
1553 3 4903 23 8539 3
1559 6 4909 47 8543 6
1567 5 4919 3 8563 2
1571 48 4931 6 8573 9
1579 12 4933 2 8581 3
1583 1 4937 1 8597 1
1597 15 4943 12 8599 18
1601 3 4951 3 8609 18
1607 9 4957 12 8623 342
1609 3 4967 16 8627 1
1613 1 4969 3 8629 2
1619 1 4973 1 8641 54
1621 17 4987 17 8647 3
1627 54 4993 3 8663 7
1637 1 4999 2 8669 13
1657 5 5003 6 8677 2
1663 5 5009 1 8681 6
1667 25 5011 2 8689 2
1669 11 5021 1 8693 9
1693 2 5023 2 8699 1554
1697 46 5039 7 8707 5
1699 2 5051 3 8713 5
1709 3 5059 60 8719 3
1721 7 5077 120 8731 2
1723 2 5081 7 8737 5
1733 1 5087 537 8741 570
1741 6 5099 1 8747 1
1747 8 5101 11 8753 1
1753 113 5107 2 8761 48
1759 2 5113 48 8779 2
1777 2 5119 3 8783 3
1783 12 5147 1 8803 2
1787 36 5153 1 8807 18
1789 14 5167 2 8819 9
1801 2 5171 483 8821 3
1811 3078 5179 2 8831 1
1823 9565 5189 4 8837 72
1831 3 5197 42 8839 5
1847 1 5209 2 8849 1
1861 2 5227 2 8861 4
1867 2 5231 1 8863 5
1871 36 5233 5 8867 3
1873 6 5237 4 8887 3
1877 6 5261 18 8893 30
1879 6 5273 3 8923 8
1889 1 5279 1 8929 33
1901 3 5281 71 8933 3
1907 4 5297 15 8941 3
1913 1 5303 16 8951 6
1931 1 5309 10 8963 702
1933 3 5323 378 8969 1
1949 34 5333 18 8971 18
1951 8 5347 6 8999 7
1973 138 5351 1 9001 57
1979 6 5381 186 9007 2
1987 3 5387 1 9011 6
1993 6 5393 7 9013 11
1997 1 5399 1 9029 60
1999 479 5407 42 9041 30
2003 40 5413 2 9043 3
2011 2 5417 6 9049 2
2017 2 5419 50 9059 1
2027 10 5431 5 9067 261
2029 311 5437 3 9091 17
2039 1 5441 4 9103 5
2053 83 5443 47 9109 24
2063 228 5449 12 9127 3
2069 1 5471 3 9133 3
2081 1 5477 426 9137 3
2083 2 5479 14 9151 72
2087 1 5483 3 9157 2
2089 6 5501 42 9161 3
2099 3 5503 107 9173 226
2111 12 5507 18 9181 8
2113 8 5519 1 9187 6
2129 1 5521 3 9199 78
2131 5 5527 5 9203 42
2137 2 5531 639 9209 1
2141 18 5557 359 9221 4
2143 2 5563 30 9227 1
2153 1 5569 2 9239 1
2161 6 5573 1 9241 2
2179 2 5581 3 9257 1
2203 2 5591 42 9277 2
2207 63 5623 2 9281 1
2213 1 5639 7 9283 156
2221 3 5641 5 9293 1
2237 4 5647 833 9311 6
2239 5 5651 1 9319 5
2243 130 5653 6 9323 43
2251 5 5657 75 9337 20
2267 7 5659 6 9341 31
2269 2 5669 1 9343 2
2273 3 5683 3 9349 3
2281 3 5689 3 9371 36
2287 120 5693 6 9377 1
2293 3 5701 5 9391 2
2297 1 5711 6 9397 17
2309 1 5717 6 9403 2
2311 2 5737 3 9413 3
2333 6078 5741 1 9419 24
2339 13 5743 5 9421 3
2341 11 5749 3 9431 6
2347 2 5779 20 9433 8
2351 3 5783 1 9437 1
2357 3 5791 2 9439 6
2371 11 5801 24 9461 19
2377 3 5807 1 9463 18
2381 192 5813 364 9467 1
2383 6 5821 5 9473 441
2389 14 5827 5 9479 9
2393 1 5839 2 9491 1
2399 6 5843 3 9497 15
2411 198 5849 24 9511 9
2417 1 5851 8 9521 4
2423 1 5857 11 9533 3
2437 3 5861 1 9539 3
2441 3 5867 31 9547 6
2447 1 5869 47 9551 1
2459 4 5879 1 9587 3
2467 6 5881 6 9601 2
2473 5 5897 19 9613 3
2477 1 5903 55 9619 672
2503 8 5923 2 9623 15
2521 5 5927 52836 9629 1
2531 1 5939 24 9631 2
2539 5 5953 35 9643 3
2543 4 5981 1 9649 11
2549 28 5987 1 9661 2
2551 18 6007 2 9677 1
2557 2 6011 9 9679 2
2579 1 6029 3 9689 24
2591 >50000 6037 17 9697 30
2593 1332 6043 5 9719 25
2609 3 6047 43 9721 2
2617 18 6053 21 9733 66
2621 1 6067 6 9739 2
2633 48 6073 12 9743 63
2647 3 6079 17 9749 36
2657 6 6089 42 9767 9
2659 2 6091 11 9769 2
2663 1 6101 4 9781 2
2671 11 6113 6 9787 5
2677 420 6121 5 9791 228
2683 9 6131 39 9803 1
2687 6 6133 5 9811 15
2689 8 6143 3 9817 5
2693 18 6151 3 9829 2
2699 3 6163 6 9833 3
2707 30 6173 1 9839 1
2711 93 6197 1 9851 3
2713 12 6199 12 9857 1
2719 12 6203 3 9859 24
2729 1 6211 74 9871 56
2731 2 6217 45 9883 18
2741 9 6221 15 9887 7
2749 5 6229 2 9901 59
2753 6 6247 5 9907 3
2767 3 6257 16 9923 4
2777 9 6263 6 9929 24
2789 16 6269 1 9931 102
2791 3 6271 3 9941 1
2797 8 6277 2687 9949 3
2801 21 6287 190 9967 2
2803 18 6299 28 9973 2
2819 6 6301 3 etc
The five asterisks indicate infinity. No finite number of 1s will ever make those primes prime again. The six greater-than symbols show lower limits for unknowns. The largest known numbers are:
74703 @ 7481
73362 @ 6701
52836 @ 5927
47850 @ 6421
46026 @ 739
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