50000 Ruth-Aaron pairs

This 5000 Ruth-Aaron-pairs addition to my compilation (50000 factored pairs @ 6.3 MB) arrived much faster than my previous updates, because two days of the calculation were done on the new, much faster iMac. From now on, therefore, I'll update only when I have collected another 10000 pairs. My current aim for the compilation is ~150000 pairs (up to ~175 billion) @ ~20 MB.

A small premium

I have run that program now for 24 hours. Whereas Mathematica was checking 1.5 billion numbers per day in the greater-than-100-billion search-range, this search has managed to look at 1.8 billion numbers in its first day (it's at 27 billion) — because factoring smaller numbers takes less time. In spite of that small premium, it'll still take until mid-March to complete the run to 100 billion. This is, nevertheless, a significant improvement over my previous estimate of August-or-so, had I not had access to the new iMac.

Double take

One of my pre-retirement, beer-store acquaintances emailed me yesterday: Are you using Mathematica 8 yet? The copies of Mathematica 4 and 5 on my old machine are Power-PC only, so I can't run them on my wife's Intel. Mathematica 6 is what is collecting the Ruth-Aaron numbers greater than 100 billion. This evening, I installed Mathematica 7 on her iMac and had it running the same program — but starting at 25.2 billion (which is where I was at). Each Mathematica runs at close to 100% CPU because the iMac's "i5" chip is dual-core. Running Mathematica 8 on top of that would only serve to degrade the performance. ;)

How fast?

I've now had my wife's new iMac collecting, in Mathematica, Ruth-Aaron numbers (greater than 100 billion) for one full week. In that period of time, it has looked at some 10.5 billion candidates — roughly one million every minute. This is more than four times the rate that I have been getting on my old Mac — in a range where the numbers are five times the size. So, I could reach 100 billion (from my current position of 24.5 billion) by mid-March (at the latest) if I were to transfer the job to my wife's iMac today. Instead, I will let her computer continue its current compilation for another two weeks.

45000 Ruth-Aaron pairs

I've added another 5000 Ruth-Aaron pairs to my list: 45000 factored pairs @ 5.7 MB.

4000555

I have now collected more than 275000 terms of A179066 and have encountered the first few values thereof exceeding 1124577 (the assumed length of this sequence) — which, therefore, lack a symmetrical counterpoint in the x=y mirror. Term #261379 is the very first of these with a value of 4000555.

Transition

It's been a long time coming but I finally got my wife to replace her "Bondi Blue" with a new "mid-2010" iMac. She insisted on the 21.5" screen but I maxed out every other spec. This will allow me to run Mathematica in the background and give me a handle on just how much faster this machine is compared to my old workhorse. I have started it collecting Ruth-Aaron numbers greater than 10^11.

My secret journal

When I started my "blahgodo" blog on 6 April 2008, I deliberately disabled its search and comment capabilities. It sat on my computer with a single link in my "chesswanks" portal, hidden in my avatar's glasses. I notified a handful of friends/acquaintances about the blog and eventually ended up with a handful of followers (who, by and large, were not the people that I had notified). The first Google mention of the word "blahgodo" did not appear until 14 August 2008.

By contrast, Glad Hobo is both searchable and commentable, but I have yet to tell a single soul about it. And although it is tempting, I will not, for I am curious just how long it will take to get that out-of-the-blue first comment.

A large Ruth-Aaron pair

A large (3109-digit) Ruth-Aaron pair from Jens Kruse Andersen in 2006:

1905497580199509189284170340160617729454507519284447380894032125401299216605231959404746406491456076087537452500615450864475294509574851694872558965147637458142847851169925504384597156839186342774703794218835079914787224201023704381910357892050633144667126701025695274917267244904046631547412451217764892682665820564602733117589573504948536227821345923192129480709161020261811449211910222044348747482379430725999743632221747352369895912918537806024983087876413015656051286630362514544336250655742275490390123627033642602816846082773618047618057241712864339666641628923304519793358493637984057056708431504856716666486852783064558823303214311031327843985329990250941661759058076234800283993403507076244375580959674432176165987722177157610949025014540262241440456225977481329826275439637821113978467713347592277272778761877369486620895052353323089466275186762154730760340318912354139790806897575696138703430821296308462103569017153398443267481786206210886767724818966146946844173334119114448743319742017799652250972014097975370944402009702825892737428030948767100797625686574016828306896510177585513807643247111028436387799038506440379364429459573363671897648038471055229959043865825572200500964733710963640202420287896153160330276281618227487298617540255476280074559350892254762502210817894443167720935846502569560506110667691599491208731919024952421363870761711803501239185538574069391983342719955717058882890896368138193494147004587331151843148675367549228967086029644320447859065120962952089596374447116233085434132873847609496753472985903393433789231694813099478361000748611580864826593011061847641838713556713471254725932258368864283683692200692371209106534770572307218871042916413593183485395365506379734347460063367332536222904080159915268050488463591666677883273886773515619453824283477025504454432123743394046125013460971239497800890371700898189508814245802605172076033911743962026904069806100895805280160105496873001315827715390516280823634234028772631187703911152987130309050986582352435463047787954537597743446653459394894961821259826264830962239473426783366979045957590674800827386475515258839150265832778437479906407032823425469516491021728001379715431899719366154512563502547285212591255838316026666455949752922243730064842786597562244544978884895727223273891803087198867866259079665555303285891331560572045662554268535966709593656589236064020830654852199844137321000857229185204802690120413827152016690707913406032458904557410391839906976201600478728787648878095918260077763451151802737157974388818693751187019698506269372899366791461115456256745354823759379125968739181772530377198570431567908347972211264615201563021468158000344014411840455964750289454658586942925561481991489057183205484114267948734129461009557038116373961305283946779805357275258052988081142338786052205011092957765100556971046168725405699196504876638805413437488325425290215402290526524610163462014214603181590049079455254375903521028239539201511806075458657292398588631028289834824303308800555229520881653153370337142941704610130531032478966096949380681751035597648875605646925429060966480912553804692548277672433233670259
and the adjacent
1905497580199509189284170340160617729454507519284447380894032125401299216605231959404746406491456076087537452500615450864475294509574851694872558965147637458142847851169925504384597156839186342774703794218835079914787224201023704381910357892050633144667126701025695274917267244904046631547412451217764892682665820564602733117589573504948536227821345923192129480709161020261811449211910222044348747482379430725999743632221747352369895912918537806024983087876413015656051286630362514544336250655742275490390123627033642602816846082773618047618057241712864339666641628923304519793358493637984057056708431504856716666486852783064558823303214311031327843985329990250941661759058076234800283993403507076244375580959674432176165987722177157610949025014540262241440456225977481329826275439637821113978467713347592277272778761877369486620895052353323089466275186762154730760340318912354139790806897575696138703430821296308462103569017153398443267481786206210886767724818966146946844173334119114448743319742017799652250972014097975370944402009702825892737428030948767100797625686574016828306896510177585513807643247111028436387799038506440379364429459573363671897648038471055229959043865825572200500964733710963640202420287896153160330276281618227487298617540255476280074559350892254762502210817894443167720935846502569560506110667691599491208731919024952421363870761711803501239185538574069391983342719955717058882890896368138193494147004587331151843148675367549228967086029644320447859065120962952089596374447116233085434132873847609496753472985903393433789231694813099478361000748611580864826593011061847641838713556713471254725932258368864283683692200692371209106534770572307218871042916413593183485395365506379734347460063367332536222904080159915268050488463591666677883273886773515619453824283477025504454432123743394046125013460971239497800890371700898189508814245802605172076033911743962026904069806100895805280160105496873001315827715390516280823634234028772631187703911152987130309050986582352435463047787954537597743446653459394894961821259826264830962239473426783366979045957590674800827386475515258839150265832778437479906407032823425469516491021728001379715431899719366154512563502547285212591255838316026666455949752922243730064842786597562244544978884895727223273891803087198867866259079665555303285891331560572045662554268535966709593656589236064020830654852199844137321000857229185204802690120413827152016690707913406032458904557410391839906976201600478728787648878095918260077763451151802737157974388818693751187019698506269372899366791461115456256745354823759379125968739181772530377198570431567908347972211264615201563021468158000344014411840455964750289454658586942925561481991489057183205484114267948734129461009557038116373961305283946779805357275258052988081142338786052205011092957765100556971046168725405699196504876638805413437488325425290215402290526524610163462014214603181590049079455254375903521028239539201511806075458657292398588631028289834824303308800555229520881653153370337142941704610130531032478966096949380681751035597648875605646925429060966480912553804692548277672433233670260

Snow plow

I wrote about our Hazel in the snow two years ago. She had a good plow this morning.

Paw flex

Thunder-Paws, demonstrating her paw flex.

40000 Ruth-Aaron pairs

I've added another 5000 Ruth-Aaron pairs to my list: 40000 factored pairs @ 5 MB.

Automobile number

An article describing Thomas A. Edison's Latest Invention appeared in Scientific American (the weekly journal of practical information) one hundred years ago today. Right beside it is a piece called The Modern Pleasure Electric Vehicle. (William Hudson writes: "The electric pleasure car is coming into its own. For certain kinds of service it is ideal. Its simple and responsive control will always be its most remarkable feature. The gasoline car has perhaps obscured the development which the electric has been undergoing in recent years.") Several other car-related articles follow in this special automobile number: Some Remarkable Mechanical Road Guides, Why You Can Buy a Good Car for Little Money, The Commercial Motor Truck vs. the Horse, A Few Shop Jobs on an Old Car, and Repainting the Old Car at Home. There follows a nice two-page spread showing twenty-five motor cars, complete with vital statistics (except for some damaged entries at the bottom of my pages). There's an article titled The Small, Inexpensive Garage. Then one about Automobile Cylinder Lubricating Oils and, on the page opposite, an advertisement for that new Edison storage battery (pictured) made by the Edison Storage Battery Company.

The big picture

I had Mathematica create a very large (4834 x 4766 pixels) graph of A179066. Most browsers will shrink the image to fit window size — which is how it looks best — but then one can expand it to actual size to better appreciate detail. The highest-up points in this part of the graph — there are higher-ups (>200000) that are not shown — are these: {97860,121125}, {98065,121114}, {98067,121134}, {98076,121143}, {98470,121123}, {98506,121132}, {98560,121141}, {98605,121177}, {98607,121152}, {98650,121213}, {98670,121215}, {98704,121222}, {98706,121224}, {98740,121231}, {98760,121233}. Their x=y mirror images are on the far right.

I had not really intended for Glad Hobo to be only recreational mathematics (as it has been so far) but rather a less-constrained, different-emphasis successor to my "blahgodo" blog. Tomorrow I will publish something different.

685/252

= 11 + 22 - 33/44*55 + 66/77 - 88/99 - 110 + 121

This is a pretty good approximation to the number e.

96319/30660

= 3*7/15 + 1/292/1 + 1/1 + 2/1/3 + 1/14

The fraction in the title approximates π, and the 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14 are the first thirteen terms of π's simple continued fraction (= 80143857/25510582).

Fractals everywhere

Perhaps it should not come as a surprise that sequence A179066 has fractal properties. I have put here two images: First, of values less than 7500, then of values less than 75000. The document has shrunk the pictures for easier comparison: Click on each one to look at the graphs full-size.

Progress and prognosis

Last night I reached term #35000 in my evaluation of A179066. I had been trying a few program improvements on yet a fourth copy of Mathematica to see if I could speed things up a little. Most of these didn't really pan out — the overhead of the new code canceling any hoped-for benefits — but one obvious implementation did shave some time off the calculations, so I aborted my run and started the new routine from where I left off. Now I am about to pass term #70000 and — putting one of my two Ruth-Aaron-pairs tabulations on hold in order to devote more CPU muscle to this — tomorrow morning I should be passing term #100000.

My expectation is that — any day now — someone else will have calculated all 1124577 terms and that will be the end of my effort. In particular, Maximilian Hasler — who co-authored A179066 with Eric Angelini — thinks that "it should not take that long".

Computing power

How difficult might it be to calculate term #1124577 of A179066? For me, very. I've been running my program now for a couple of days and have only managed some 25000 terms. I can probably see my way to 100000 terms but, short of a significantly more optimized version of my search routine (and even with that, the search beyond would prove taxing), there it will end. Remember also that I am running two other copies of Mathematica in the background (my ongoing Ruth-Aaron pairs tabulation) and these will eat up available computer-clock cycles, reducing the Mathematica kernel's CPU usage to some 60% of the available 2 MHz. Have I mentioned yet that I could really use a modern replacement for my seven-and-a-half-year-old machine?

Proof by contradiction

Yesterday I asked for the "final term" of a sequence. It may not be immediately obvious that there is for this sequence such a thing, but consider this: A179066 is defined in such a way that term #1124578 must have a digital root of 1 (the digital root of 1124578) and yet be composed of only the digits 0, 3, 6, and 9 (those digits that are not used in 1124578). But numbers made up of only 0s, 3s, 6s, and 9s will be evenly divisible by 3 or, alternatively, leave a remainder of 0, 3, or 6 when divided by 9 (making their digital roots 3, 6, or 9), thus contradicting our already established fact that the number that we are looking for has a digital root of 1. As defined, term #1124578 cannot exist and term #1124577, therefore, is likely the final term of this sequence.

28, 11, 12, 13, ...

Eric Angelini came up with this: Each successive term is the smallest positive integer not yet used, where both the term and its index have the same digital root (remainder when divided by 9) but do not have in common any of each other's base-ten digits. What might be this sequence's final term?

35000 Ruth-Aaron pairs

I've just added another 5000 Ruth-Aaron pairs to my ongoing tabulation thereof (bringing the total now to 35000 pairs). The project will likely consume the better part of this year. I realize that this would go much, much faster using a modern computer (my dual 2 GHz PowerPC G5 is now well over seven years old) but, in the absence of anyone else taking this on, at least I've got a decent start on it.

1+2+3/45/6/7*89

Using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in order, replace each comma with one of +, -, *, /, or a concatenation of adjacent numbers. Parentheses or other punctuation are not allowed and, as usual, multiplications and divisions are to be evaluated before additions and subtractions. My title today evaluates to a quantity fairly close to π. Can you come up with an expression that is even closer?

Countdown

Just how likely was my pre-midnight countdown? Consider the ordering of 10, 9, 8, 7, 6, 5, 4, 3, 2, 1 and allow the commas to be replaced by one of +, -, *, /, or a concatenation of adjacent numbers (with the added proviso of no concatenation after the 10 to force the generally accepted start of the countdown). We exclude parentheses, so the mathematical statements must be evaluated in the standard order of operations (i.e., multiplications and divisions evaluated before additions and subtractions). Of the 1562500 (4*5^8) possible numerical outcomes, 486501 are unique. Of these, only 51249 are integers (31373 positive), the largest being 10*987654321. The smallest positive integers that do not appear are 1992, 2102, 2133, etc., so every year from 1993 to 2101 will have at least one solution: The largest number of solutions in this interval for a given year is 13 and the average, 616/109. Only two of the solutions start with 10/ (1997 = 10/9*87*6*5 - 43*21 and 2021 = 10/9*8*765/4 + 321). While 1992 is missing from our integer solutions, it may be approximated by the nearby 10 + 9 + 8 + 76*543/21 (as well, if we allow concatenation after the 10, it's 1098 + 765 + 4*32 + 1). You might like to look for one of the four solutions that evaluate to 377/120, the closest that we can get to the irrational number π.

An interesting bonus in last night's countdown exemplar is the symmetry of the operations: +, *, *, /, *, *, +. How common might that be? Out of the 616 solutions in the 1993-2101 interval, I count 12 that qualify:

1994 = 10*9 - 8*7 + 654*3 - 2*1
2001 = 10 - 9*87 + 65*43 - 21
2011 = 10 + 9*8*7/6*5*4 + 321
2020 = 10 - 9 + 8*7 + 654*3 + 2 - 1
2023 = 10*987/6 + 54/3*21
2024 = 10 + 9 + 8*7*6*5 + 4 + 321
2040 = 10 + 9 + 8*7 + 654*3 + 2 + 1
2072 = 10 + 9*8 + 7 + 654*3 + 21
2088 = 10*9*87/6/5/4*32*1
2093 = 10*98 + 7*6*5 + 43*21
2098 = 10 - 98*7 + 65*43 - 21 or 10 - 9*8 + 7 - 6 + 5*432 - 1