In February I explained the status of my current Leyland-primes search, hoping therein to have scored my first Leyland prime with more than 100000 decimal digits by this summer.
Today I achieved that milestone with 23406^18781+18781^23406, which contains in its expanded form 100031 decimal digits.
Tuesday, April 17
Saturday, April 14
Lost in conversion
I thought I'd give the Netflix reboot of Lost in Space a go just now. To be clear, I'm old enough to have watched the mid-1960s original and I didn't think much of it. Three minutes into the first episode of the remake, the starship is losing altitude:
I've enabled the subtitles in the screen grab so that you might comprehend my objection. I decided not to watch any further.
I've enabled the subtitles in the screen grab so that you might comprehend my objection. I decided not to watch any further.
Tuesday, April 10
Ancestry
I got my ancestry-by-populations chart from 23andMe today:
No real surprises. My father's family is the German component. The Netherlands and Denmark matches just serve to regionalize which part of Germany (northwest) that family lived. Poland is there, of course, because that was my mother's mother's place of birth. We had always thought of my mother's father as Russian and the match for Ukraine is a nominal step in that direction, as one mustn't take these countries literally.
Somewhat more surprising is the 8.7% British and Irish component. It doesn't mean, I'm sure, that I have an ancestor that hails therefrom. Rather, some British and Irish populations share a common ancestry with me.
No real surprises. My father's family is the German component. The Netherlands and Denmark matches just serve to regionalize which part of Germany (northwest) that family lived. Poland is there, of course, because that was my mother's mother's place of birth. We had always thought of my mother's father as Russian and the match for Ukraine is a nominal step in that direction, as one mustn't take these countries literally.
Somewhat more surprising is the 8.7% British and Irish component. It doesn't mean, I'm sure, that I have an ancestor that hails therefrom. Rather, some British and Irish populations share a common ancestry with me.
Wednesday, April 4
California dreamin'
Yesterday morning I noticed a couple of small plastic STUCCO advertising-signs nailed to the hydro utility pole in front of my house and — sure enough — two more on a pole up the street. I've been waging war on these signs for a good number of years! So I surveyed the immediate neighborhood (some six streets) and found two more pairs, on Lippincott St. W. and Edmund Ave. In the afternoon — wearing my high-visibility safety vest, a hammer under my belt and a small ladder on my shoulder — I took them all down.
In my survey I discovered a "medical marijuana" dispensary on the northwest corner of Weston Rd. and Lippincott St. W. (The only time I might find myself at this intersection is for the across-the-street Dairy Queen, which is closed in the winter.)
This is the first such business (of which I am aware) in Weston, albeit the south end thereof. Apparently, it is a concern.
In my survey I discovered a "medical marijuana" dispensary on the northwest corner of Weston Rd. and Lippincott St. W. (The only time I might find myself at this intersection is for the across-the-street Dairy Queen, which is closed in the winter.)
This is the first such business (of which I am aware) in Weston, albeit the south end thereof. Apparently, it is a concern.
Saturday, March 31
Go fund me
I wrote about our local "reverend" Patrick White of the Chaplain's Office here and here. When I was doing all that research on him I somehow missed the February 2017 fundraiser set up by York St. Peter's, an evangelistic organization based in Woodbridge ON. White's friend, Oliver, is noted as being in "serious condition" after he "was overcome with smoke and suffered 3rd degree burns" fighting a stove fire in White's apartment — which Oliver had been assigned to "watch" while White was out of town. Oliver, I guess, was augmenting his watching with a little cooking.
Another casualty of the fire was Patrick White's parakeet, Joey, who came through the ordeal with a bit of smoke inhalation. Joey is mentioned in a photo on the Chaplain's Office website:
Wait a second! Your friend Oliver was seriously hurt in a fire in your apartment and all you can say is that he was recently married ?!
Another casualty of the fire was Patrick White's parakeet, Joey, who came through the ordeal with a bit of smoke inhalation. Joey is mentioned in a photo on the Chaplain's Office website:
Wait a second! Your friend Oliver was seriously hurt in a fire in your apartment and all you can say is that he was recently married ?!
Thursday, March 15
More English number words in base-26 pi
I last worked on this in January 2013. Dropping the final "googol" line from that post's in situ list, but continuing with subsequent finds:
These are the numbers' first occurrences, at indices 10087, 11324, 13463, 14295, 15276, 64838, 175372, 389247, 786244, 1556763, 2300987, 8879098, 9202330, 9946442, 33027856, 126003234, 126348794, 238426469, 389952198, 536531272, 1709539474, 5624040208, 7690604024, 7869864133, respectively.
I excluded "googol" because it's just a slang name for "ten duotrigintillion". Likewise, I've excluded "hundred", "million", "billion", because in proper English I expect them to be modified by a numerical adjective. Regardless, for the record, here they are:
... whose shown first occurrences are at indices 454315613, 942420517, 2169343694, 4359384810, respectively. In December 2000, I created a sequence of numbers: 1, 4, 2, 6, 10, 1, 2, 2, 6, 10, 6, 10, 2, 2, 5, 2, ... representing the (proper) number words in the order in which they occur. I now have a very large number of terms for this sequence. Back then I noted eight consecutive 6s by showing them in blue. In my now-larger list, there are ten consecutive 6s. There are also eleven consecutive 2s, twelve consecutive 1s, and thirteen consecutive 10s.
..datnmvbwezlfygflonebugwcjooatbjbskh..
..wzlwdhgbhphxeqsqfourdjjmmikzqwvivabm..
..pceoeiigwyqtmaustwoamxyatxhqxoyzbno..
..ifnuetcfdxrwipodsixyavsnasweexagllo..
..ectasqgcghjmzirrtenfjiphdkdoumqzpjt..
..dwbfgegkqjwkubjvfivewgjlbihzmwgalifm..
..jyhdeevmsxmbxtbininesuzrnrzsxosctuxm..
..efzoqitnhuljsgztzerovlptejdfwretaara..
..hwdvoynebabsyjkdsevenaabgjthpqgapqnrg..
..cgakthjgxsmoldokthreeapdiuhncqcjgsmnt..
..gfaoforthfmdhgkzfiftyxiboudqllqzqlcmh..
..obptjczwqdirtnvmfortyigfxwmybvztjkhbc..
..hubksmltneixkvpheightmgsxrwaqokrdcpnw..
..ttsybfashnuibvlisixtytdchrjcvvzazzivu..
..wltescttnomsztonelevenfyifyclxdcacsfta..
..qpddtdyuitfganmgninetyhecteviwajvcosrv..
..kvvtnvsdypwsumzveightysstqdeajfbyxrrde..
..iuwxlmueqjjzpbcmtwelveefcsszipcnmifmzp..
..ssoshdnbsnembdkgthirtyqwbfxohiadwgtvya..
..fcgcgwvbsyihoavxtwentymbigcbkjfiuezgex..
..edapsvilnlndkwusfifteenzkfkepmwhsauydgc..
..mkczsxsgslvltfxqnineteenpxzlphhysfrfpjwp..
..gmrxvlhvuocxdssaseventyurdrqcdywvowytwb..
..qndvrurhkjaqjqabsixteennwdbnngckhpzduwr..
These are the numbers' first occurrences, at indices 10087, 11324, 13463, 14295, 15276, 64838, 175372, 389247, 786244, 1556763, 2300987, 8879098, 9202330, 9946442, 33027856, 126003234, 126348794, 238426469, 389952198, 536531272, 1709539474, 5624040208, 7690604024, 7869864133, respectively.
I excluded "googol" because it's just a slang name for "ten duotrigintillion". Likewise, I've excluded "hundred", "million", "billion", because in proper English I expect them to be modified by a numerical adjective. Regardless, for the record, here they are:
..dnihymgxncwxfhsdgoogolxvodlfjhbzlnrucr..
..knqqklhgazusrbummillionziaocmauzjxqqrdo..
..bcqtdtvmnmumvmhqhundrednizjohkrhvlzdkpv..
..qgrqqgrlwjuobeyfbillionliuisnrlcobglhiv..
... whose shown first occurrences are at indices 454315613, 942420517, 2169343694, 4359384810, respectively. In December 2000, I created a sequence of numbers: 1, 4, 2, 6, 10, 1, 2, 2, 6, 10, 6, 10, 2, 2, 5, 2, ... representing the (proper) number words in the order in which they occur. I now have a very large number of terms for this sequence. Back then I noted eight consecutive 6s by showing them in blue. In my now-larger list, there are ten consecutive 6s. There are also eleven consecutive 2s, twelve consecutive 1s, and thirteen consecutive 10s.
Tuesday, March 6
Facts first
This local red-light camera setup became operational on 20 October 2017. The first photo shows the scene on the northeast corner of Lawrence Ave. W. and Weston Rd. A second picture shows it from the other direction:
A reporter for WestonWeb (an acquaintance to whom I still owe a couple of beers) seemed to be unaware of its existence when he suggested that "there are only 77 red light cameras in the whole of Toronto and only one remotely close to our area (at Keele and Lawrence)". His erroneous 77 was apparently based on this City of Toronto website:
Banana, banana, banana! It wasn't too difficult to extract the actual apple from the basket. The supervisor of Toronto's Red Light Camera Operations assured me that — as of today — there are 124 red-light cameras in operation.
Tuesday, February 27
Saturday, February 24
The long compute
Around the middle of 2017, I started looking for Leyland primes between L(40210,287) and L(40945,328). L(x,y) = x^y + y^x and we assume x >= y > 1. L(40210,287) is Leyland #324766365 and L(40945,328) is Leyland #349812824. So there are 25046458 Leyland numbers between them which I want to check for probable primality. Any given check may take a significant amount of time since we are dealing with numbers that are roughly 100000 decimal digits long.
I haven't been totally committed to the task for the entire period but I may perhaps have spent six months on it. Since I've only covered about one fifth of the territory, I have two years to go! I was going to add some processors to the task but my intended purchase of a new machine has (sadly) been stymied. There was a second issue. My list of sorted consecutive Leyland numbers only went up to #331682621, having been computed with the sole objective of reaching 100000-digit numbers (which it did). I thought I was going to need the new computer to calculate more terms because my indexing computation was limited by available RAM and my current machines can't take any more than 64 GB. Fortunately, I recently discovered that that was sufficient to extend the indexing to L(40945,328).
The good news is that I have so far found eight previously unknown Leyland primes, ranging in size from 98889 to 99659 decimal digits. By this summer I should have scored my first Leyland prime with more than 100000 decimal digits. There are currently only nine known Leyland primes <decimal digits> larger than this:
I haven't been totally committed to the task for the entire period but I may perhaps have spent six months on it. Since I've only covered about one fifth of the territory, I have two years to go! I was going to add some processors to the task but my intended purchase of a new machine has (sadly) been stymied. There was a second issue. My list of sorted consecutive Leyland numbers only went up to #331682621, having been computed with the sole objective of reaching 100000-digit numbers (which it did). I thought I was going to need the new computer to calculate more terms because my indexing computation was limited by available RAM and my current machines can't take any more than 64 GB. Fortunately, I recently discovered that that was sufficient to extend the indexing to L(40945,328).
The good news is that I have so far found eight previously unknown Leyland primes, ranging in size from 98889 to 99659 decimal digits. By this summer I should have scored my first Leyland prime with more than 100000 decimal digits. There are currently only nine known Leyland primes <decimal digits> larger than this:
L(40945,328) <103013> Norbert Schneider Dec 2014
L(41507,322) <104094> Norbert Schneider Dec 2014
L(222748,3) <106278> Anatoly Selevich Dec 2010
L(45405,286) <111532> Norbert Schneider Apr 2015
L(48694,317) <121787> Norbert Schneider Aug 2015
L(234178,9) <223463> Anatoly Selevich Jul 2011
L(255426,11) <265999> Serge Batalov May 2014
L(314738,9) <300337> Anatoly Selevich Feb 2011
L(328574,15) <386434> Serge Batalov May 2014
Wednesday, January 24
Pi-primes
3, 31, 314159, and 31415926535897932384626433832795028841, are the first four terms in the "pi-primes" sequence: A005042. That 38-digit fourth term was attributed by Martin Gardner (in 1979) to Robert Baillie and Marvin Wunderlich. By 2000, a larger (fifth) term had yet to be found. That year, Clifford Pickover (under the guise of Dr. Googol) wrote in his "Wonders of Numbers" that there are likely infinitely many terms but "neither humans nor any lifeforms in the vast universe will ever know the next prime... It is simply too large for our computers to find." I wrote about Pickover's gross underestimation of computational progress previously and this serves as another example.
In 2001, Ed T. Prothro calculated that fifth term, composed of 16208 digits. In 2006, Eric Weisstein calculated the sixth and seventh terms, composed of 47577 and 78073 digits, respectively. In 2016, Adrian Bondrescu calculated the eighth term, composed of 613373 digits.
A fine point is that — as of this writing — the fifth to eighth terms are not proven primes, but probable only. That should not deter one from (pragmatically) thinking of them as primes.
In 2001, Ed T. Prothro calculated that fifth term, composed of 16208 digits. In 2006, Eric Weisstein calculated the sixth and seventh terms, composed of 47577 and 78073 digits, respectively. In 2016, Adrian Bondrescu calculated the eighth term, composed of 613373 digits.
A fine point is that — as of this writing — the fifth to eighth terms are not proven primes, but probable only. That should not deter one from (pragmatically) thinking of them as primes.
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