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.

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:

I attempted to get him to acknowledge that the number was small but he didn't think that that was too important: "... whatever the actual number and location is, it’s inadequate" and then doubled down on the number: "I believe that unmarked red light cameras should be at many more intersections than the current 77."

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


The blue mesh consists of unit squares. What is the area of the black rectangle?

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:

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


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.

Saturday, January 13

Ice jam

Yesterday morning, a mild spell had fractured enough of the local Humber river ice — created over several weeks of bitter cold — to have started a small ice "jam". The ice jumbles reach river bottom and thus prevent water from flowing underneath. Diverted water flows in the Raymore Park floodplain on the other side. Here's how it looked from there:

At noon, the ice backup reached ~300 meters ...

... but an hour later, chunks of new upstream ice were pushing into it:

Here are a couple more views from Raymore Park:

Somewhat surprisingly, the Raymore island beaver was out and about:

By this morning, the river was flowing once again. Some of the ice had become wedged in the narrow channel on the east side of Raymore island:

I believe that this year's ice jam is the first significant one in seven years. My record of the 2010 ice jam (with a link to the 2009 one) is here.