Happy New Year!

## Thursday, December 19, 2019

### True probabilities in self-referential statements

On December 8, รric Angelini asked in an online MathFun forum for a solution to a self-referential sentence in which randomly picking four letters from the statement yielding the four letters F, O, U, and R, was a probability ๐ถ/๐ท, where ๐ถ and ๐ท were the English number names for the numeric quantities that allowed the sentence to be true. After determining that picking one letter depleted the letter availability of the next pick by one, I settled on my template being: "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are ๐ถ out of ๐ท."

Mathematica has a function for converting integers into English words so all I had to do was run through a bunch of them and test the resulting sentences for truthfulness. Here are nine solutions:

1. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are two out of two hundred nineteen thousand six hundred eighty-seven."

2. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of two hundred ninety-two thousand nine hundred sixteen."

3. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are three out of one hundred ninety thousand six hundred fifty."

4. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are six out of seven hundred twenty-one thousand seven hundred ninety-one."

5. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are twelve out of one million seven hundred sixty-six thousand six hundred twenty-two."

6. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are eighteen out of one million six hundred thousand two hundred."

7. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are twenty-one out of two million ninety-seven thousand twenty-four."

8. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are thirty-five out of two million six hundred sixty-seven thousand four hundred twenty."

9. "The odds of randomly picking four letters from this statement and having them be F, O, U, and R, are seventy-one out of nine million nine hundred twenty thousand two hundred sixty-two."

I believe that this exhausts the possibilities for ๐ถ up to 100 and ๐ท up to 10000000. But only for this particular template. One may easily alter the template by adding/subtracting/changing words without altering the essential thrust of what the sentence is saying. Each of these new statements would have its own set of solutions.

Of the above nine solutions, the first two are special: The ratio ๐ถ/๐ท is in its lowest terms. In the other seven the two numbers share a common factor, so they can be reduced. Of course we mustn't do that because that would rob the statements of their truthfulness.

If you are interested in verifying the nine statements, this will help:

1.  {2,219687}  {5,11,6,8}  <132>   5*11*6*8/(132*131*130*129) = 2/219687
2.  {3,292916}  {5,11,6,9}  <132>   5*11*6*9/(132*131*130*129) = 3/292916
3.  {3,190650}  {7,10,6,9}  <126>   7*10*6*9/(126*125*124*123) = 1/63550
4.  {6,721791}  {5,11,6,8}  <135>   5*11*6*8/(135*134*133*132) = 2/240597
5. {12,1766622} {5,12,6,8}  <145>   5*12*6*8/(145*144*143*142) = 2/294437
6. {18,1600200} {5,12,6,8}  <128>   5*12*6*8/(128*127*126*125) = 1/88900
7. {21,2097024} {6,13,5,7}  <130>   6*13*5*7/(130*129*128*127) = 7/699008
8. {35,2667420} {7,12,7,10} <147>  7*12*7*10/(147*146*145*144) = 1/76212
9. {71,9920262} {5,13,6,8}  <146>   5*13*6*8/(146*145*144*143) = 1/139722

After {๐ถ๐ท} are the letter counts of {F,O,U,R}, followed by the statement's <total-letter-count>, followed by the probability calculation and ๐ถ/๐ท in its lowest terms.

## Friday, December 13, 2019

### Sapphire

Today is the sixth Friday, December 13th anniversary of our marriage. The photograph of the somber newlyweds was taken by my friend, Alfy Marcuzzi.

## Thursday, December 12, 2019

### A330365

The idea for OEIS sequence A330365 came from รric Angelini who asked me if I was willing to check and extend it. I was happy to. On the OEIS page for the sequence one can click "graph" and get (at the bottom) a logarithmic scatterplot. I have a much prettier version:

Click on the picture for a larger display of it. The blue points are the values at odd indices; the red points, values at even indices. Green lines connect adjacent values.

## Wednesday, November 20, 2019

### 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.

## Friday, October 25, 2019

### Revenant gems

รric Angelini's revenant numbers became OEIS sequence A328095. In รric'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

## Friday, September 27, 2019

### 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).

## Tuesday, August 27, 2019

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, 2019

### 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.

## Friday, August 23, 2019

### 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, 2019

### 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, 2019

### 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!

## Saturday, August 10, 2019

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   Jan 10
6  L(30247,300) - L(40089,82)    8561240  <75828>   38       Feb 23   Feb 22
7  L(40089,82)  - L(40746,91)   15146841  <78282>   69       May  2   May 22
8  L(40746,91)  - L(32160,329)   5639471  <80390>   27       May 29  (May 22)
9  L(32160,329) - L(40495,114)  11887307  <82129>   57       Jul 25  [Jul 11]
10  L(39070,143) - L(91382,9)    15717090  <85712>   78       Oct 11  [Aug  6]
11  L(91382,9)   - L(35829,302)   8886580  <88031>   46       Nov 26  [Aug 14]
12  L(35829,302) - L(37738,243)   6370928  <89444>   33       Dec 29  [Aug 20]
13  L(37738,243) - L(38030,249)   6038222  <90579>   32  2021 Jan 30  [Aug 26]
14  L(37614,265) - L(40210,287)  43838597  <95032>  241       Sep 28  [Sep 19, 2020!]

#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.

## Thursday, August 01, 2019

### 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 09, 2019

### 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, 2019

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 >100000
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 >100000           3491 36                6983 1
401 1                 3499 18                6991 44
409 2                 3511 8                 6997 2
419 24                3517 >100000           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 >100000           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 >100000           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 >100000           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 > symbols indicate solutions (if any) greater than 100000. The largest known numbers are:

74703 @ 7481
73362 @ 6701
52836 @ 5927
47850 @ 6421
46026 @  739