Thursday, October 01, 2020

On target (revisited)

one hundred eight
one hundred nine
one hundred ninety-eight
one hundred ninety-nine
two hundred forty
two hundred forty-one
two hundred forty-three
two hundred forty-four
two hundred forty-five
two hundred forty-six
two hundred forty-seven
two hundred forty-eight
two hundred forty-nine
two hundred fifty
two hundred fifty-one
four hundred fifty-three
four hundred fifty-four
five hundred fifty-nine
one thousand, one hundred seventy-four
one thousand, seven hundred sixteen
five thousand, five hundred fifty-six
five thousand, five hundred fifty-seven
six thousand, nine hundred fifty-six
six thousand, nine hundred fifty-seven
fifteen thousand, seven hundred fifty-six
seventeen thousand, one hundred fifty-five
twenty-four thousand, nine hundred ninety-eight
twenty-four thousand, nine hundred ninety-nine
forty-three thousand, five hundred sixty-eight
forty-three thousand, five hundred sixty-nine
seven hundred thirty-five thousand, seven hundred fifty-nine
one million, one hundred five thousand, eight hundred five
one million, one hundred five thousand, eight hundred six
one million, one hundred five thousand, eight hundred seven
one million, one hundred seven thousand, seven hundred eighty-four
one million, one hundred seven thousand, seven hundred eighty-five
one million, five hundred eighty-four thousand, five hundred three
one million, five hundred eighty-four thousand, five hundred four
one million, seven hundred seven thousand, nine hundred forty
one million, seven hundred seven thousand, nine hundred forty-one
one million, nine hundred twenty-one thousand, five hundred sixty-six
one million, nine hundred twenty-one thousand, five hundred sixty-seven
two million, two hundred seventeen thousand, seven hundred nine
two million, two hundred seventeen thousand, seven hundred ten
two million, five hundred sixty-eight thousand, seven hundred one
two million, five hundred sixty-eight thousand, seven hundred two
three million, five hundred forty-one thousand, two hundred seventy-nine
three million, five hundred forty-one thousand, two hundred eighty
three million, five hundred forty-one thousand, two hundred eighty-one
three million, five hundred forty-one thousand, two hundred eighty-two
three million, five hundred forty-one thousand, two hundred eighty-three
three million, five hundred forty-one thousand, two hundred eighty-four
three million, five hundred forty-one thousand, two hundred eighty-five
four million, five hundred twelve thousand, five hundred ninety-six
four million, five hundred twelve thousand, five hundred ninety-seven
twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-seven
twenty-eight million, two hundred sixty-three thousand, eight hundred twenty-eight

Monday, September 28, 2020


The red polygon atop a Google map (click on it for a better view) represents the 15 hectare confines of my time in south Weston / north Mount Dennis since March 20, with the sole exception of a trip to the vet on June 10. Yesterday, having finally received some cloth masks via Amazon, I decided to break out and walked to the yellow open-circle spot to the northwest, where I took this photo of a condo construction site:

Monday, September 21, 2020

Root words (revisited)

A few days ago, Futility Closet highlighted a David Morice "Kickshaws" bit (from a 1997 "Word Ways") dealing with "root words" wherein the number of letters of a specific power of an integer written in English words is equal to that integer. This looked like something I could verify and possibly expand on. In short order, Mathematica came up with over one hundred examples. I trust of course that Mathematica is correctly providing the (American) English equivalents of the large numbers. Also, Mathematica does not ever include the word "and" in its integer wording. The solutions list begins:

3^0 = 1: one
4^1 = 4: four
9^2 = 81: eighty-one
10^2 = 100: one hundred
10^6 = 1000000: one million
10^7 = 10000000: ten million
10^9 = 1000000000: one billion
10^10 = 10000000000: ten billion
34^3 = 39304: thirty-nine thousand, three hundred four
50^7 = 781250000000: seven hundred eighty-one billion, two hundred fifty million
57^5 = 601692057: six hundred one million, six hundred ninety-two thousand, fifty-seven
60^9 = 10077696000000000: ten quadrillion, seventy-seven trillion, six hundred ninety-six billion
66^4 = 18974736: eighteen million, nine hundred seventy-four thousand, seven hundred thirty-six
70^8 = 576480100000000: five hundred seventy-six trillion, four hundred eighty billion, one hundred million
82^6 = 304006671424: three hundred four billion, six million, six hundred seventy-one thousand, four hundred twenty-four
84^5 = 4182119424: four billion, one hundred eighty-two million, one hundred nineteen thousand, four hundred twenty-four
88^5 = 5277319168: five billion, two hundred seventy-seven million, three hundred nineteen thousand, one hundred sixty-eight
88^6 = 464404086784: four hundred sixty-four billion, four hundred four million, eighty-six thousand, seven hundred eighty-four

In addition to the five powers of ten, 88 has both a fifth and a sixth power that qualify. Another integer with multiple powers (59 and 60) is 2326. My final entry (the full list is here):

3411^83 = 169505077051870754738656270298647848404856613833009765867153972878059706815231408155486012246885256492743555141825746171932166403032624146125802858179156124954911215966912803668763183796090700586141654608753614304795644365897817582455004645907480837144225881607352543409279188776048394771808331: one hundred sixty-nine senonagintillion, five hundred five quinonagintillion, seventy-seven quattuornonagintillion, fifty-one trenonagintillion, eight hundred seventy duononagintillion, seven hundred fifty-four unonagintillion, seven hundred thirty-eight nonagintillion, six hundred fifty-six novoctogintillion, two hundred seventy octoctogintillion, two hundred ninety-eight septenoctogintillion, six hundred forty-seven sexoctogintillion, eight hundred forty-eight quintoctogintillion, four hundred four quattuoroctogintillion, eight hundred fifty-six tresoctogintillion, six hundred thirteen duooctogintillion, eight hundred thirty-three unoctogintillion, nine octogintillion, seven hundred sixty-five novemseptuagintillion, eight hundred sixty-seven octoseptuagintillion, one hundred fifty-three septenseptuagintillion, nine hundred seventy-two seseptuagintillion, eight hundred seventy-eight quinseptuagintillion, fifty-nine quattuorseptuagintillion, seven hundred six treseptuagintillion, eight hundred fifteen duoseptuagintillion, two hundred thirty-one unseptuagintillion, four hundred eight septuagintillion, one hundred fifty-five novemsexagintillion, four hundred eighty-six octosexagintillion, twelve septensexagintillion, two hundred forty-six sesexagintillion, eight hundred eighty-five quinsexagintillion, two hundred fifty-six quattuorsexagintillion, four hundred ninety-two tresexagintillion, seven hundred forty-three duosexagintillion, five hundred fifty-five unsexagintillion, one hundred forty-one sexagintillion, eight hundred twenty-five novemquinquagintillion, seven hundred forty-six octoquinquagintillion, one hundred seventy-one septenquinquagintillion, nine hundred thirty-two sexquinquagintillion, one hundred sixty-six quinquinquagintillion, four hundred three quattuorquinquagintillion, thirty-two tresquinquagintillion, six hundred twenty-four duoquinquagintillion, one hundred forty-six unquinquagintillion, one hundred twenty-five quinquagintillion, eight hundred two novemquadragintillion, eight hundred fifty-eight octoquadragintillion, one hundred seventy-nine septenquadragintillion, one hundred fifty-six sexquadragintillion, one hundred twenty-four quinquadragintillion, nine hundred fifty-four quattuorquadragintillion, nine hundred eleven tresquadragintillion, two hundred fifteen duoquadragintillion, nine hundred sixty-six unquadragintillion, nine hundred twelve quadragintillion, eight hundred three novemtrigintillion, six hundred sixty-eight octotrigintillion, seven hundred sixty-three septrigintillion, one hundred eighty-three sextrigintillion, seven hundred ninety-six quintrigintillion, ninety quattuortrigintillion, seven hundred trestrigintillion, five hundred eighty-six duotrigintillion, one hundred forty-one untrigintillion, six hundred fifty-four trigintillion, six hundred eight novemvigintillion, seven hundred fifty-three octovigintillion, six hundred fourteen septenvigintillion, three hundred four sexvigintillion, seven hundred ninety-five quinvigintillion, six hundred forty-four quattuorvigintillion, three hundred sixty-five trevigintillion, eight hundred ninety-seven duovigintillion, eight hundred seventeen unvigintillion, five hundred eighty-two vigintillion, four hundred fifty-five novemdecillion, four octodecillion, six hundred forty-five septendecillion, nine hundred seven sexdecillion, four hundred eighty quindecillion, eight hundred thirty-seven quattuordecillion, one hundred forty-four tredecillion, two hundred twenty-five duodecillion, eight hundred eighty-one undecillion, six hundred seven decillion, three hundred fifty-two nonillion, five hundred forty-three octillion, four hundred nine septillion, two hundred seventy-nine sextillion, one hundred eighty-eight quintillion, seven hundred seventy-six quadrillion, forty-eight trillion, three hundred ninety-four billion, seven hundred seventy-one million, eight hundred eight thousand, three hundred thirty-one

This is likely the largest solution you will see for a while. Mathematica's large-integer naming system is relatively recent and when it was introduced a few years ago it arrived with a bug. Although that "-illions" bug was eventually fixed, the naming of integers 10^306 and larger remains essentially aberrant/unusable. If you are curious about how many times each letter appears in the preceding:

{a,68}, {b,1}, {c,29}, {d,209}, {e,369}, {f,86}, {g,113}, {h,148}, {i,441}, {l,195}, {m,8}, {n,426}, {o,238}, {p,19}, {q,50}, {r,184}, {s,108}, {t,307}, {u,197}, {v,69}, {w,22}, {x,50}, {y,74}.

There are, in addition, 371 spaces, 97 commas, and 70 hyphens. If you are a user of Mathematica, I will alert you to the fact that the hyphen it uses for its integer names is perversely different from the keyboard hyphen that I have used here.

Sunday, September 06, 2020

My largest Leyland prime find

Late yesterday, I found my (to-date) largest Leyland prime: 33845^26604+26604^33845. At 149763 decimal digits, this becomes now the fifth largest known such prime:

386434  (328574,15)      Serge Batalov        May 2014
300337  (314738,9)       Anatoly Selevich     Feb 2011
265999  (255426,11)      Serge Batalov        May 2014
223463  (234178,9)       Anatoly Selevich     Jul 2011
149763   (33845,26604)   Hans Havermann       Sep 2020

The number is the 167th new Leyland prime discovered since I (using xyyxsieve and pfgw) started finding them two months ago. Prior to that I had found 579 new Leyland primes using Mathematica — but that took from 3 October 2015 to 3 July 2020. At my current rate of discovery, I will find my 1000th new Leyland prime on December 9, but that is likely early because I am entering large-number terrain where my finds will be slower in coming. Still, I might have it by the end of the calendar year. We'll see.

Saturday, September 05, 2020

Visually impaired

Going behind the garden shed just prior to sunset, August 25, I had a smallish raccoon approach me atop the neighbour's back fence. It seemed not to notice me until fairly close, whereupon it retreated to that yard's maple tree. I ran in the house to fetch my camera. It was only after looking at the photos just now that I noticed the eyes.

Display bug


I have had insects walk on my iMac display screen before but this one is behind the glass.

Sunday, August 23, 2020

The iMac from hell

I had mentioned in my previous post that I was considering purchasing a new computer "if I can trade in my old kernel-panic-plagued late-2015 one". When I bought this iMac it was pretty much top of the line.

SSDs were expensive and 1 TB was as big as Apple was willing to provide at the time. The standard 8 GB RAM was going to be replaced with 64 GB purchased from Other World Computing (Apple has always been incredibly ungenerous with its memory pricing).

I experienced my first kernel panic on the new iMac on 23 December 2015, 15 days after receipt. There was another one on 9 February 2016. I seem to have been panic-free until May 8 and then more on May 14, May 21, and May 22, when I decided to do a RAM switch. In addition to the kernel panics there was other weirdness happening in my running Mathematica Leyland prime searches. For example, the Mathematica front end would lose its connection with the Mathematica kernel. Mathematica was pretty much the only thing I was running on this machine.

I tried an OS reinstall on June 13 but the situation did not improve. I should of course have brought it back to Apple before its one-year warranty ran out but I was still under the impression that it might be the third party RAM and, besides, Catherine was increasingly anxious about driving and I did not ask her for the ride.

By July 2017 I had boxed the iMac from hell and purchased a new one, the one that I am currently working on. It's been fine but the 2 TB SSD is getting full. A 4 TB SSD in a new iMac should do the trick for a few more years. Plus I can upgrade to 128 GB RAM. The idea of trading in the defective iMac seemed like a good way to reduce the cost.

Phobio is Apple's official Mac Trade In partner for U.S and Canada. When I tried to evaluate nine days ago my late-2015 iMac on Phobio's website (by entering the computer's serial number), I got this:

My device (iMac Core i7 4.0 GHz) did not appear! Selecting one of the Core i5 alternatives instead would of course end up in my iMac being undervalued and this was not fair. So I sent an email (August 14) to Phobio explaining my dilemma. No response. A followup email (August 19) noting the non-response and asking for them to "please let me know" has also not gotten a reply. At this point I am entertaining deep-Apple conspiracy theories about what is going on (and I'm not the sort of person who entertains conspiracy theories).

Strangely, my late-2014 iMac which has never had kernel panics before started to experience them in June 2019. At any rate, I will give up for the moment the idea of trading in the late-2015 machine. I'll likely still purchase a new iMac but am unsure as to when.

Monday, August 10, 2020

A look ahead (reprise)

A year ago today I charted the expected progress on my five-year indexing-the-Leyland-primes project. Having a few days ago finished interval #10, I anticipate intervals #11 to #13 to be done by September, and #14 by October, all thanks to Mark Rodenkirch's xyyxsieve and pfgw.

my current work sheet

As I am no longer burdened by my previous reliance on Mathematica 9, I may even update everything to macOS Catalina one of these days. A new iMac is also being considered (if I can trade in my old kernel-panic-plagued late-2015 one). After I've finished interval #14, I will have tabulated all Leyland primes up to 103013 digits. Interval #15 will start there and continue on to the end of interval #28, reaching 150000 digits:

15  L(40945,328) - L(41507,322)   6612071   105334 (5e9)  6  Aug 28 - Sep 30
16  L(41507,322) - L(222748,3)   13527824   217348 (5e9) 12  Sep  3 - Oct 10 ~
17  L(222748,3)  - L(45405,286)  33460389   536426 (5e9)
18  L(45405,286) - L(48694,317)  69041008
19  L(48694,317) - L(44541,746)  43871809
20  L(44541,746) - L(49205,532)  45659518
21  L(49205,532) - L(49413,580)  18377349   287809 (e10)
22  L(49413,580) - L(49878,755)  54608684
23  L(49878,755) - L(144999,10)  11614904   182243 (e10)
24  L(144999,10) - L(145999,10)   8050111   126465 (e10)
25  L(145999,10) - L(146999,10)   8094919   127396 (e10)
26  L(146999,10) - L(147999,10)   8139747   128441 (e10)
27  L(147999,10) - L(148999,10)   8184494   128015 (e10) 12  Sep 27 - Nov  7 ~
28  L(148999,10) - L(149999,10)   8229120   129812 (e10) 12  Aug 31 - Oct 11 ~

The column after the interval is the number of Leyland numbers between the end-points of that interval, followed by how many of those remain after sieving to the subsequent quantity (in brackets). This is followed by the number of iMac-mini cores working on primality testing and the start and finish dates (~ added if in the future). Initially, in addition to intervals #15 and #16, I will be having a go at interval #28 in order to up my PRPTop production score.

Tuesday, July 21, 2020

An interesting prime sequence

The sequence starts 2, 3, 11, 23, 29, 61, 19, 113, 157, 127, 103, ... These are the red endings of the following (second column) triangular array. The first column numbers are the indices. The third column numbers are the sums of the digits of the second column integers.

There are several constraints imposed in creating our sequence. Each successive term must be a distinct prime. It must be the smallest such prime that allows the following: At index k, take the final k digits of the sequence. The first of those final k digits must not be zero in order that we may have the concatenation of those digits be a k-digit prime (these are our middle numbers). Finally, the sum of those k digits (the third column) must also be a prime.

  1  2  2
  2  23  5
  3  311  5
  4  1123  7
  5  12329  17
  6  232961  23
  7  3296119  31
  8  96119113  31
  9  119113157  29
 10  9113157127  37
 11  13157127103  31
 12  315712710343  37
 13  5712710343149  47
 14  71271034314989  59
 15  271034314989701  59
 16  7103431498970113  61
 17  10343149897011341  59
 18  343149897011341751  71
 19  3149897011341751379  83
 20  49897011341751379499  101
 21  897011341751379499373  101
 22  9701134175137949937397  109
 23  11341751379499373971223  101
 24  341751379499373971223293  113
 25  1751379499373971223293601  113
 26  13794993739712232936011471  113
 27  949937397122329360114711303  109
 28  9937397122329360114711303223  103
 29  73971223293601147113032231297  101
 30  712232936011471130322312973547  101
 31  2232936011471130322312973547619  109
 32  32936011471130322312973547619769  127
 33  936011471130322312973547619769683  139
 34  6011471130322312973547619769683433  137
 35  11471130322312973547619769683433503  139
 36  471130322312973547619769683433503563  151
 37  7113032231297354761976968343350356337  157
 38  13032231297354761976968343350356337239  163
 39  322312973547619769683433503563372395333  173
 40  2312973547619769683433503563372395333337  181
 41  12973547619769683433503563372395333337311  181
 42  297354761976968343350356337239533333731147  191
 43  9735476197696834335035633723953333373114771  197
 44  35476197696834335035633723953333373114771673  197
 45  761976968343350356337239533333731147716731889  211
 46  9769683433503563372395333337311477167318895801  211
 47  69683433503563372395333337311477167318895801211  199
 48  683433503563372395333337311477167318895801211409  197
 49  3433503563372395333337311477167318895801211409277  199
 50  33503563372395333337311477167318895801211409277313  199
 51  356337239533333731147716731889580121140927731311003  193
 52  3372395333337311477167318895801211409277313110031607  193
 53  23953333373114771673188958012114092773131100316071109  191
 54  953333373114771673188958012114092773131100316071109281  197
 55  3333731147716731889580121140927731311003160711092811381  193
 56  33731147716731889580121140927731311003160711092811381613  197
 57  311477167318895801211409277313110031607110928113816133361  197
 58  4771673188958012114092773131100316071109281138161333611103  197
 59  71673188958012114092773131100316071109281138161333611103263  197
 60  731889580121140927731311003160711092811381613336111032631283  197
 61  8895801211409277313110031607110928113816133361110326312834153  199
 62  58012114092773131100316071109281138161333611103263128341531697  197
 63  121140927731311003160711092811381613336111032631283415316971039  197
 64  1140927731311003160711092811381613336111032631283415316971039179  211
 65  40927731311003160711092811381613336111032631283415316971039179347  223
 66  277313110031607110928113816133361110326312834153169710391793472971  229
 67  7313110031607110928113816133361110326312834153169710391793472971151  227
 68  13110031607110928113816133361110326312834153169710391793472971151349  233
 69  100316071109281138161333611103263128341531697103917934729711513492351  239
 70  3160711092811381613336111032631283415316971039179347297115134923513943  257
 71  60711092811381613336111032631283415316971039179347297115134923513943541  263
 72  110928113816133361110326312834153169710391793472971151349235139435411459  269
 73  9281138161333611103263128341531697103917934729711513492351394354114592617  283
 74  81138161333611103263128341531697103917934729711513492351394354114592617137  283
 75  138161333611103263128341531697103917934729711513492351394354114592617137937  293

I'm indebted to √Čric Angelini for the seed of the idea.