Monday, December 26, 2016

Lateral thinking

Today's Futility Closet features lateral thinking — which folk of my generation associate with Edward de Bono's 1967 "The Use of Lateral Thinking". Wikipedia suggested that de Bono had coined the expression. After a quick check with Google Books, I decided to replace "coined" with "promulgated" but, alas, most of my earlier-than-1967 references turned out to be later ones — a hazard of Google Books' date-assignment methodology.

Fortunately, I did find one that did not so evaporate. It is from Edward T. Owen's article "Linguistic Aberrations" in the "Transactions of the Wisconsin Academy of Sciences, Arts, and Letters" (volume XXIII, 1927, page 434):

In matters mental, for convenience, let it be supposed that we have reached our limit with e. g. the idea-plexus indicated by the woman and her prepossessing qualities. Our words for these we shall marshal as we please, for the ideas of each of us are simultaneous. The like is true of the husband and his qualities. And even when I say "There is a handsome _ _ _ _ woman with an ugly _ _ _ _ husband", I exhibit what might also be regarded as a single mental picture, since it is continuous. Its continuity is however that of the old fashioned panorama. Its constituent pictures come upon the mental stage successively. No doubt, indeed, while thinking of the woman I am also conscious of the man to come. The mind is somewhat like the eye; even the momentary scope of either is considerable. While intently looking at you I take note of the surrounding objects. While I add my bank deposits, I am wondering if the total will be big enough to pay my bills. The eye does much by lateral vision and the mind by lateral thinking.* But such seeing and such thinking are both vague. My lateral thinking of the man to come is not presumably distinct enough e. g. to meet the needs of word-selection.
________
* I do not intend by "lateral" that only which is figuratively conceivable as to the right or left. I have no faith in mental limitations thus suggested. Were the word "eccentric" not preempted, it would serve my purpose of suggesting what is in the mind but is not central — whether to the right or left, above, below or even too near, too far — say out of focus — I care little.

Friday, December 09, 2016

Earthquakes

The U.S. Geological Survey has a latest earthquakes website that you can configure (and then, bookmark) by clicking on the white gear icon (top right) for options and settings. From the defaults I change the Earthquakes to 1 Day, All Magnitudes Worldwide, the Map Layers to Terrain, and the Time Zone to Local System Time. You can then click on the now-cyan gear icon to dismiss the right settings panel. Likewise, you can click on the cyan list icon to dismiss the left list panel. That leaves you with a cyan globe icon and a default map of the United States. You can now zoom out (or in) by using the plus/minus signs at the top left (or by using your scroll wheel). I get it to some manner of world-view that fits my particular browser window size and this I bookmark.

When a large earthquake hits, I scroll wheel over the place to zoom into it and often leave the browser tab there for a few days to watch the aftershocks develop. The following is a screen grab of the current situation based on yesterday's 7.8 Solomon Islands quake. With Auto Update on, new aftershocks will appear and stay red for an hour.


I love the detail in the zoomed-in maps. I'm intrigued by the fact that (in the above map of Makira) four of the islands to its north have an identical high point of 91 m (above sea level).

Friday, December 02, 2016

As it happens


When you hear/see a helicopter hovering overhead here in Toronto, it's an indication of a nearby news story.

Friday, November 25, 2016

A deep walk in base-two pi

After I managed to calculate 8 billion ternary digits of π, I figured that I could probably generate 10 billion binary digits thereof. My previous effort was 1 billion binary digits. I won't bother graphing the walk but I will report that my output of the indices of zero therein has jumped from 45915 to 147043.

Addendum: I've stepped through those 10 billion digits a couple more times. In the walk, I counted 5738590822 terms that are positive and 4261262135 terms that are negative. I've also checked record maxima and minima and submitted them as two new sequences: A278737 and A278738.

A deep walk in base-three pi

In my previous post I set up a graphical depiction of the ternary digits of π by "walking" them from the origin to (950,2678) in 25 million steps. Here is an alternate view — in orange — of the same walk (click on the picture to better see the detail):


This is a compressed version of my previous graph as only one of every one thousand points is shown and connected. I have added an x=y line to indicate where along the walk the accumulated number of ones and twos might be equal. (950,2678) is shown as a black dot. Starting at that point — in cyan — I have continued the walk (a bit of the left and a lot of the right of it has been cropped). The full walk to 8 billion looks like this:


A hint of orange near the origin betrays the change in scale of the continuation. After step 8 billion we have reached (42488,102078), also marked by a black dot. Which is to say that we have a digit deficit at this point of 102078 zeros and 59590 twos! I counted 96619 times that the number of ones and twos were exactly equal. And we are left with 0, 15, and 18 initial ternary-π digits as the only known instances where the number of all three digits are exactly equal.

Addendum: I have upped the walk from 8 billion steps to 10 billion steps, where we have reached (35427,93367).

Wednesday, November 16, 2016

A walk in base-three pi

Looking at the output of my previous post, it would be natural to believe that A039624 is infinite. What about a base-three analogue? For how many of the initial digits in the ternary representation of π are there solutions where the number of zeros, ones, and twos are all exactly equal? In base three, π = 10.010211012222010211... After a preliminary no-digits, the first 15 and 18 digits also clearly qualify. Then what?

I wanted to explore this graphically, so I converted π's ternary digits to vectors in a Cartesian coordinate system. I let the digit 1 be the vector (0,1); the digit 2, (1,0); and the digit 0, (-1,-1). If we start at (0,0), every time we revisit the origin we should have an exactly equal number of zeros, ones, and twos:

 1    (0,0)  +  (0,1)  =  (0,1)
 2    (0,1)  + (-1,-1) = (-1,0)
 3   (-1,0)  + (-1,-1) = (-2,-1)
 4   (-2,-1) +  (0,1)  = (-2,0)
 5   (-2,0)  + (-1,-1) = (-3,-1)
 6   (-3,-1) +  (1,0)  = (-2,-1)
 7   (-2,-1) +  (0,1)  = (-2,0)
 8   (-2,0)  +  (0,1)  = (-2,1)
 9   (-2,1)  + (-1,-1) = (-3,0)
10   (-3,0)  +  (0,1)  = (-3,1)
11   (-3,1)  +  (1,0)  = (-2,1)
12   (-2,1)  +  (1,0)  = (-1,1)
13   (-1,1)  +  (1,0)  =  (0,1)
14    (0,1)  +  (1,0)  =  (1,1)
15    (1,1)  + (-1,-1) =  (0,0)
16    (0,0)  +  (0,1)  =  (0,1)
17    (0,1)  + (-1,-1) = (-1,0)
18   (-1,0)  +  (1,0)  =  (0,0)
19    (0,0)  +  (0,1)  =  (0,1)
20    (0,1)  +  (0,1)  =  (0,2)
...

And indeed, we re-arrive at the origin at steps 15 and 18. Then what?


The graph shows 25 million steps, with the initial (0,0) and final (950,2678) points shown in red. Yes, we should get back to the origin eventually but it's not as easy to believe as it is for base-two. It should be more understandable then that providing an exactly-equal number of all base-b digits in some initial terms of base-b π with b > 3 will be difficult. Regardless, we have in base-four, the first 4 digits (3.021) and in base-five, the first 75 digits!

A walk in base-two pi

In base two, π = 11.001001000011111101101010100010001... Ignoring the decimal point and replacing every zero with minus one, we get: 1, 1, -1, -1, 1, -1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, -1, 1, ... We can now treat these numbers as a one-dimensional walk on the y-axis. Accumulating: 1, 2, 1, 0, 1, 0, -1, 0, -1, -2, -3, -4, -3, -2, -1, 0, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 0, 1, 0, -1, -2, -1, ... Finally, we graph this sequence:


The picture is very much compressed, showing only one of every one hundred points. What looks like a single crossing from positive to negative territory at 10^8, plotting the values from 100000000 to 101400000 details to this:


Even this detail is lacking. What appear to be a couple of dozen y = 0 values are actually 2409 such, ranging from 100023378 to 101375384. The list of zeros constitute the terms of A039624 and I've just calculated 45915 of them.

Monday, November 14, 2016

Fear of height


The Humber river retaining wall is approaching its northern limit. In the photo (above), the house at the top-right is actually at the end of the street on which I live. Yesterday, I climbed to the top of the wall at its southern end — five meters above the (shallow) water. Here are two pictures taken from that vantage — looking across to the other side of the river, and towards the north:



 It didn't feel the least bit safe and I was glad to climb down again.

Saturday, November 05, 2016

More 3-balanced factorization integers

Four weeks ago I discovered what I now know to be the smallest base-ten 3-balanced factorization integer. By the end of October I had completed my base-ten k-balanced list to 10^12 and decided to extend the search to 1.5 * 10^12, thus including (when finished) the first of the four 13-digit 2-balanced factorization integers. I was of course also hoping for more (small) 3-balanced examples. Yesterday and today one of my processors found these (which thusly become the second-, third-, and fourth-smallest):

1045675984884 = 2^2 * 3 * 17 * 67 * 103 * 359 * 2069
1046959786860 = 2^2 * 3 * 5 * 13 * 43 * 2087 * 14957
1047697856460 = 2^2 * 3^2 * 5 * 109 * 347 * 153889

My ill-fated, eight digits reverse search is now back on track, though I realize that it will take a good while to complete. It will rediscover the twenty-one 12-digit 2-balanced factorization integers and — more excitingly — has already come up with a k-balanced solution for k > 2.

Sunday, October 30, 2016

Weston's John Street pedestrian bridge


On Friday, the John Street UP/GO rail crossover was, at long last, opened to the public. Historically, this location may be considered to be the heart of Weston because it is where its train stations (CNR, CPR) used to be. Architecturally, I thought the bridge to be quite nice.


Although the safety fencing gets in the way of on-the-bridge photography, I was able to find a lower platform that provided an unobstructed view of the new train station to the south:


Sunday, October 09, 2016

A small 3-balanced factorization integer

My eight digits reverse search has run into unexpected and unwelcome difficulties. I have put it on a back burner for now. The forward search to 10^12 continues on several fronts and, today, snagged on one of those fronts a 12-digit 3-balanced factorization integer! In my mind I had conjectured a 13-digit 3-balanced factorization integer but a 12-digit find came as a total surprise. This answers my query as to the existence of a 3-balanced integer smaller than the largest 2-balanced integer:

785984586660 = 2^2 * 3^2 * 5 * 7 * 13 * 149 * 307 * 1049

Wednesday, September 28, 2016

So long, Teksavvy

On 17 September I experienced a significant outage on my Teksavvy internet feed. When it finally came back, I rebooted the modem, noticed that my domain no longer pointed to my computer, and made the change to the new IP address at DynDNS. I then sent an email message to an acquaintance. I have an automatic BCC to myself to let me know that messages are in fact being sent — and this one was not. So I tried some more test messages, none of which were being delivered!

Two days later, the situation had not resolved itself and I Gmailed Teksavvy support for help. After the usual security clearance, they sent me a long questionnaire — meant to rule out some obvious causes of failure. Some of the directions in that questionnaire made it obvious that they had no more than a passing acquaintance with Apple OS X and, along with my answers, I expressed: "I'm a little concerned about your lack of Mac experience as evidenced by several mistakes in using-the-terminal instructions. Other OS X references are stale at best. The current OS X is 10.12 and the location of the 'Network Utility' hasn't been where you say it is for some two years!"

On 26 September, they finally opened a ticket with their "Tier 2" department. This meant, I suppose, that this was not an obvious failure and they were prepared to take my issue seriously. What followed was an attempt to have me send a message in something other than Apple Mail because they were able to send something (using my account) by way of Microsoft Outlook. And I did manage eventually to send something using Mozilla Thunderbird, so it was (obviously) only a problem with the Apple Mail program.

Well, not so fast. On 25 September I had created an email account with Bell. The setup was straightforward and my first message — using Apple Mail — arrived at its destination immediately. Nothing wrong with Apple Mail! From Teksavvy's perspective, all I had to do to fix my problem was ditch Apple Mail. From my perspective, that was never going to happen. I've been using Apple Mail for some fourteen years. I've been using Teksavvy for five years. So long.

Wednesday, September 21, 2016

13-digit 2-balanced factorization integers

When I introduced my k-balanced factorizations last month, I noted that there were over 13000 of the 2-balanced variety. My effort to chart them all is proceeding slowly, with a couple of months yet to go before I reach 10^12. In the meantime I decided to have a try at the ones greater than 10^12. This can be accomplished by brute-forcing all possible ways of creating a valid factorization using seven or fewer digits. I generated no solutions for one to four digits. For five digits, one gets the 4 solutions of the known 1-balanced factorizations (that started it all):

        26487 = 3^5 * 109
        28651 = 7 * 4093
        61054 = 2 * 7^3 * 89
        65821 = 7 * 9403

For six digits, there are again no solutions. For seven digits, there are 4 solutions:

1495476527089 = 83^2 * 601^3
3392164558027 = 7^9 * 84061
8789650571264 = 2^31 * 4093
9418623046875 = 3^9 * 5^10 * 7^2

These then are (necessarily) the largest 2-balanced factorization integers! I'm going to run the program again for eight digits. This will generate all 12-digit 2-balanced factorization integers and may do so (I'm hoping) in less time than my exhaustive forward search. In addition, there's a chance that it may find a k > 2 solution, since nothing in the reverse search precludes it.

Monday, September 19, 2016

Eastern milk


The only snakes I have ever seen here in Toronto are garters. So when I came across this deceased (note the ants) juvenile by the curb in the road, I had to look up what kind it was. I have since been told that an adult eastern milk snake was spotted by a homeowner across the street from this scene. That person's home is right above the new retaining wall on the Humber river. So this snake's habitat has been disrupted by the construction, resulting in its being noticed where it might otherwise never have ventured. Hopefully it survives.

Thursday, September 15, 2016

Curb your enthusiasm


The sealant in the expansion joint between a sidewalk-proper and its curb degrades over time, allowing a few hardy species to gain a foothold. I assume there's a root system that manages to find its way beneath the concrete. Gotta love these plants!

Sunday, August 28, 2016

A factorization balancing act

A couple of weeks ago, Claudio Meller presented 26487 and 65821 as examples of the property of having one each of the base-ten digits when combined with the digits of their respective factorizations. Surprisingly, he missed two:

    26487 = 3^5 * 109
    28651 = 7 * 4093
    61054 = 2 * 7^3 * 89
    65821 = 7 * 9403

I wondered how this might be turned into a sequence. Base-ten k-balanced factorization integers: The combined digits of an integer and its factorization primes and exponents contain exactly k copies of each of the ten digits. So,

 45849660 = 2^2 * 3 * 5 * 19 * 37 * 1087
 84568740 = 2^2 * 3 * 5 * 67 * 109 * 193
104086845 = 3^2 * 5 * 19 * 23 * 67 * 79
106978404 = 2^2 * 3 * 13 * 685759

and so on. For any given k, k-balanced integers are necessarily finite. For k=2, there are a little over 13000. Is the largest of these greater than the smallest 3-balanced integer?

It's not too difficult to generate very large terms:

345655692176023955231233047798293565182493067678373538829683596019374251932061880049408412541410264671864570007449201127

is an example of a 13-balanced integer. Can you come up with a larger one?

Saturday, August 06, 2016

My 100th Leyland prime find


Last October I found my first previously unknown Leyland (probable) prime. Today I found my 100th. The graph shows (in order of discovery) the number of decimal digits of those 100 primes, ranging from 43633 to 61184. The finds have had the unintended (but certainly not unpleasant) consequence of pushing me up a list of probable prime contributors! My search for Leyland primes is of course in aid of my Leyland prime indexing effort which has reached #1137. There'll be a significant jump in about three weeks when I finish rounding up the few remaining unknowns with decimal digit lengths of between 54334 and 55390.

Saturday, July 30, 2016

Point of entry


We have a totally fenced yard at the back and sides of the house. To be more accurate, a monster shed on the adjacent property at the back of the yard takes the place of a fence. The front fencing on the (gravel) driveway side accommodates a gate that can be swivelled to allow vehicles into the yard. It is this gate that is being compromised.

For a number of years now some creature has been creating a summer underpass here to allow entry into (and presumably exit from) the yard. I have a shovel nearby to refill the hole and have come to place some small concrete chunks on the yard side of the gate to give the visitor a nasty surprise half-way through its dig. Sometimes a new underpass is generated to one side of the concrete but more generally the creature gives up.

I had always thought the offender was a skunk but some weeks ago Bodie and I were surprised — in broad daylight — by a rat running through the yard, only to disappear (after it was surprised by us) under that back shed!

Friday, July 29, 2016

Flipped


Not too far from our home, this car-crash situation stopped Weston Road traffic both ways for hours. Not seen in the photo are two additional cars to the left, one of which seems to have sustained significant damage. It is difficult for me to imagine a scenario that would flip a car just so but there it is. Notice a woman's shoe sitting on the undercarriage. I haven't found a hint of this event in the media!

Thursday, July 28, 2016

Farmers' market


The long-running Weston farmers' market has moved a little closer to my home this year because of impending construction at the old (parking lot) site. I rarely go but decided to accompany Catherine (green hat, far left, below) back on the 16th. It's much smaller than the old site and — in terms of local produce or decent deals — a little disappointing.

Monday, June 20, 2016

Cubic Lock


These are the four pieces of Cubic Lock by Goh Pit Khiam as realized in exotic woods by Brian Menold. A union of the two pieces on the left (top, the key made up of 10 cubies; bottom, 9 cubies) is inserted into a union of the two pieces on the right (top, 23 cubies; bottom, 19 cubies) and by means of a strategic to-and-fro of the key, the pieces are shifted into place. It's not obvious that the finished 4*4*4 cube has an internal void until one counts and adds those cubies. A good show-off puzzle as one is unlikely to forget the assembly once one has put it together a few times.

Wednesday, June 08, 2016

The wall


I went to Raymore Park (on the other side of the Humber river) last Friday to see what progress had been made on the erosion-control retaining-wall they were putting in place (on my side of the river). Contrary to the intelligence in my previous entry on this, I can now see that the wall will not be so much "on top of the now-in-the-river foundation" as I had supposed but (rather) much higher and more-closely hugging the slope bedrock — which is actually being exposed for a more stable conglomeration. And that storm drain interruption will end up being a barrier to my ever walking along the full length of the wall. At any rate (depending on the wall-top width), it may be too high to be safe. The bits of white floating through the air (more evident in the second photo) is tree fluff.

Monday, May 23, 2016

577, 5569, ...

This incipient-sequence comment in T.D. Noe's A138290 recently caught my eye. So, prime p such that 2^(p+1)-2^q-1 is composite for all positive q < p. Carlos Rivera had it published as Puzzle 437 in 2008. Therein, Giovanni Resta notes that he had checked up to 15373 without finding a third term. All of my processors were busy of course but I hadn't been using Catherine's computer in a while (too slow for my needs), so...

This morning (after several weeks of number crunching) I noticed that her machine had come up with the third term: 29251. To give you an idea of the computing involved: on my latest iMac, p = 577 (174-digit numbers) verified compositeness for all terms immediately; p = 5569 (1677-digit numbers) took a minute; and p = 29251 (8806-digit numbers) needed more than four hours!

[The sequence became A278740 on 27 Nov 2016.]

Monday, May 09, 2016

Mount Dennis


The part of Toronto in which I live used to be a town called Weston. I live in the south part of Weston, just northwest of a community called Mount Dennis. This photograph was taken in the very northern part thereof, looking down Weston Road towards Toronto's CN Tower. My camera is at maximum zoom, giving the intersection at Jane Street (200 meters away) and the streetscape beyond a seriously foreshortened look. The Tower is 10.8 km distant. I was on my way to Jane's Walk.

Thursday, April 21, 2016

Humber river retaining wall

When I speculated one month ago that the current armourstone toe of the valley slope on the east side of the Humber river (not far from my home) was going to be replaced with a higher version, I wasn't entirely correct. It may end up being higher but (more importantly) it will also be further into the river.


Roy Murray continues his Raymore Park updates on the construction (March 30, April 12, April 18, and a follow-up). In the last of these he lets on that the retaining wall will end up on top of the now-in-the-river foundation! This is actually good news for my side of the river because it means that I might be able to access this new stretch of real estate from the south (behind Denison Park) and walk it up to the footbridge (across the river) that lies beyond. Of course Roy worries that the now-narrower river channel will negatively impact his side of the river during flood events. Here's a bird's-eye Apple Maps view of the area (I've put a red square around my house):

Saturday, April 09, 2016

8613, ...

In numbers of the form 2^n-2^m-1, m<n-1, for what values of n is the sum of the values of m for which that number is prime equal to n?

Tuesday, April 05, 2016

She said, he said

On page 20 of the 23 November 1939 Cleveland Plain Dealer is a Good Morning from Claire MacMurray column called "Thanksgiving Nightmare" which tells the tale of a "Mrs. Amos Pinchot" writing a bit of verse whilst in a dream state:

Hogamus Higamus
Men are Polygamous
Higamus Hogamus
Women Monogamous

The men/women being polygamous/monogamous was certainly nothing new. A Google Books search finds references going back into the 1800s. What was new are the two words hogamus and higamus. Who coined them?

The Reader's Digest reprinted the poem and its origin in its May 1940 issue, giving the nonsense words wide circulation. Any attribution to folk after this date is therefore moot. No one appears to have come out of the woodwork to say: Hey, that's my misappropriated poetry! But we do have something else — someone suggesting that Claire MacMurray lied about the poem's authorship.

In 1942 James Grier Miller published a book called "Unconsciousness" in which he retells the story of the writing of the verse — but from the perspective of a man! The switch was accomplished by means of a footnote stating: "Mrs. Amos Pinchot has repeatedly been incorrectly said to have been the author of this quatrain. She denies any responsibility for it, however, and the true author appears to be shrouded in anonymity." Mr. Miller was certainly no stranger to referencing assertions, so it's somewhat disconcerting to see him not do so here.

There were two women who could have claimed to having been Mrs. Amos Pinchot: Gertrude Minturn (who married Amos in 1900) and Ruth Pickering (who married him in 1919). Gertrude died in May 1939 — six months before Claire MacMurray's story appeared. If Ruth Pickering denied authorship of the poem, when and where?

It strikes me as disingenuous to hold Claire MacMurray to a standard that we do not demand of James Grier Miller. Either of them could have promulgated an untruth. [A third possibility is that Ruth Pickering — in denying authorship — did so. A fourth possibility is that Claire MacMurray was referring to Gertrude Minturn.] Yet James Miller's unsupported claim somehow demeans that of Claire MacMurray to an extent that Garson O'Toole calls her story "fanciful". No matter. In the realm of word/phrase origins, early citation is everything. If one is at all uneasy about crediting Mrs. Amos Pinchot for the creation of those fanciful words, attribute them instead to Claire MacMurray*!

* "Claire MacMurray" was her pen name. She was born Bessie Claire McMurray on 12 Feb 1899 in Huntington IN. In 1916 she swam from Liverpool to Havana (Illinois) in under four hours. Marrying Edward Howard II in 1923, they had three sons (born 1925, 1927, and 1929). Claire saw publication of her starting-in-1936 newspaper columns as two books (1941, 1944, subsequently amalgamated into a third). The first of those books inspired the 1941 NBC radio series "We're Five in the Family". Claire died 31 Jul 2003 in Cleveland OH.

Sunday, April 03, 2016

L'Anse aux Meadows to Stormy Point

Here's a dozen distance/direction quotes culled from the online stories:

National Geographic (Mar 31): "hundreds of miles south of"
New York Times (Mar 31): "about 300 miles south of"
Huffington Post (Apr 1): "about 300 miles further south than"
Daily Mail (Apr 1): "400 miles (643km) south west of"
Canada Journal (Apr 1): "300 miles (480km) south of"
Gossip Monthly (Apr 1): "about 600 km south of"
CTV News (Apr 1): "approximately 500 kilometres from"
The Telegram (Apr 1): "almost 500 km south of"
The Register (Apr 1): "300km away from"
Digital Journal (Apr 1): "about 500 kilometres (300 miles) south of"
Canadian Press (Apr 1): "about 600 kilometres from"
The Japan Times (Apr 2): "about 700 kilometers (430 miles) away"

To give credit to Digital Journal's Karen Graham, she added "on the south-west coast". My measurement in Google Earth is 501.13 km at a heading of 214.96°, which corresponds (roughly) to southwest by south (SW by S = 213.75°).

Wikipedia now sports an entry on Point Rosee: "approximately 600 kilometres (370 mi) south of L'Anse aux Meadows". I'm curious how long it will take before it mentions Stormy Point.

Saturday, April 02, 2016

Stormy Point, Millville NL


When the (possible) new Vikings-settlement story broke on Thursday it was given a dateline location of Point Rosee. This dealt amateur geographers a good measure of grief because Point Rosie (by Rosée Harbour, in Fortune Bay) is (by way of English corruption) actually Point Enragée, on the Burin peninsula — nowhere near the archaeological site.

It took a little time to determine the site's exact location. And having that, I learned that it was actually called Stormy Point. So from whence comes the "Point Rosee"? It could be a local name. However, it's possible — likely even — that the archaeologists decided to disguise the location so as to keep the public at bay.*

I have to say, Stormy Point strikes me as a poor location for a settlement. It's totally open to the elements!

* Addendum: I just finished watching the Nova episode on this. Indeed, early in the program they called the location "secret".

Thursday, March 31, 2016

Cyclopean primes

After watching the Numberphile Glitch Primes and Cyclops Numbers video yesterday, I thought it a bit of a waste (because of an overly narrow definition) that there should only be one binary Cyclops number that is prime. What if we allow the solitary zero to be anywhere (except of course leftmost) in a binary all-ones number? Well, we would have the terms in Antti Karttunen's A095078, which I'm now calling Cyclopean primes. It would be easy to generate ten or twenty thousand of these and call it a day, but I'm interested in — given that these numbers are of the form 2^n -2^m -1 with m< n-1 — how many primes (values of m) there are for any given n. So, my table starts like this (indices n, bracketed m's):

 1
 2   {0}
 3   {1}
 4   {2,1}
 5   {3,1}
 6   {4,2,1}
 7
 8   {6,5,4,2}
 9   {7,5,3,1}
10   {5,2,1}
11   {3}
etc.

There are 38218 solutions up to (and including) n=10000, so an average of 3.8 solutions for a given n. The so-far largest number of solutions is 20, occurring at n=2850 and n=9510.

Wednesday, March 30, 2016

Near-rep-nine palindromic primes

On Sunday, I gave two large (Darren Bedwell) examples of numbers in A088284. That sequence has been languishing for over ten years with only eight terms! So here (culled from Makoto Kamada's webpages) are forty more:

 1 10^3-8*10^1-1
 2 10^3-7*10^1-1
 3 10^11-8*10^5-1
 4 10^17-7*10^8-1
 5 10^19-7*10^9-1
 6 10^27-8*10^13-1
 7 10^29-5*10^14-1
 8 10^45-5*10^22-1
 9 10^53-10^26-1
10 10^73-5*10^36-1
11 10^87-8*10^43-1
12 10^177-4*10^88-1
13 10^209-5*10^104-1
14 10^225-4*10^112-1
15 10^237-2*10^118-1
16 10^339-8*10^169-1
17 10^363-8*10^181-1
18 10^397-4*10^198-1
19 10^705-7*10^352-1
20 10^757-10^378-1
21 10^1061-7*10^530-1
22 10^1245-4*10^622-1
23 10^1395-7*10^697-1
24 10^2273-5*10^1136-1
25 10^2493-10^1246-1
26 10^2631-7*10^1315-1
27 10^3159-8*10^1579-1
28 10^3597-10^1798-1
29 10^3837-7*10^1918-1
30 10^5749-7*10^2874-1
31 10^5835-10^2917-1
32 10^8457-4*10^4228-1
33 10^11753-7*10^5876-1
34 10^13537-7*10^6768-1
35 10^20105-4*10^10052-1
36 10^35729-5*10^17864-1
37 10^36155-8*10^18077-1
38 10^45305-8*10^22652-1
39 10^46069-10^23034-1
40 10^50897-5*10^25448-1
41 10^95019-10^47509-1
42 10^104281-10^52140-1
43 10^111725-4*10^55862-1
44 10^125877-7*10^62938-1
45 10^134809-10^67404-1
46 10^269479-7*10^134739-1
47 10^290253-2*10^145126-1
48 10^314727-8*10^157363-1

I cannot guarantee that the last three (Darren Bedwell terms) are contiguous with the first 45 because Kamada's stated search range for that second power of ten is only up to 68000.

I've also now watched the Numberphile Glitch Primes and Cyclops Numbers video. Glitch primes are the not-quite-palindromic near-rep-nine numbers with which I started Sunday's post. Just prior to defining the (strictly base-two) Cyclops numbers (@ 9:33) we are told that 10^19-10^9-1 is a prime! Looking at my above table, 10^53-10^26-1 should have been the example used.

Monday, March 28, 2016

Tree enumerators


Yesterday morning I spotted a couple of fellows walking on people's front properties — apparently to measure tree widths. Really, who works on Easter Sunday? Their vehicle (which had no license plate in front, Quebec license plate in back) was left idling, even when the driver was well away from it. All of this made me very suspicious!

Sunday, March 27, 2016

Shiver in ecstasy


Cliff Pickover's voluminous twitter feed includes entries introduced with "shiver in ecstasy". This one is from last Friday. The hard part in verifying the claim is counting the nines. I see an eight with 252 nines in front and 253 nines behind, so 10^506-10^253-1. It's not quite a palindrome and it really isn't that large, but we can (courtesy of Darren Bedwell) fix that:

999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999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