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 for the moment the fifth largest known such prime:

386434  (328574,15)      Sergey Batalov       May 2014
300337  (314738,9)       Anatoly Selevich     Feb 2011
265999  (255426,11)      Sergey 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.