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27 October 2005

Ramanujan's Taxicab!

Ron Bizer of the USA State Shaped Like a Human Right Hand Mitten Palm Toward Viewer -- and I must state his Full Name Here because this is a credit for Intellectual Property:

© 2005 Ron Bizer, All Rights Reserved

-- is my Old Army Buddy, and a Very Talented Artist, whom I seem to have accidentally contaminated with the Postal Art / Mail Art Virus.

He began The Life Artistic as a teenager drawing Stuff for the Rubes (that's us) at Carnivals, and told me what Rebesak and Blue Steam are. (Blue Steam is conveyed around the Midway in buckets.)

Still a teenager, he was one of a stable of small underpaid slaves whom Big Daddy Roth forced to draw a very famous series of Monsters Driving Crazy Hot Rods called Odd Rods.

THIS WONDERFUL THING
just received is an
ORIGINAL PIECE OF MAIL ART
FOR VLEEPTRON
.


It is not Hardy's Taxicab, but RAMANUJAN'S TAXICAB.

The self-taught Indian mystic mathematician Ramanujan has hailed a taxicab to drive him to Infinity.

Pizza for Ron!

No Pizza for all of you, who have still not provided (in accord with the Vleeptron PizzaQ No Googling Honor System) the Number of Hardy's Taxicab -- "the smallest number which can be expressed as the sum of two cubes in two different ways."

14 Comments:

Anonymous Jim Olson said...

Ok. I needed some time to do this one since I am a math dunce. But, here's my idea.

The question is to find the smallest number that can be expressed as the sum of two cubes, in two different ways.

So, first, I did this. (Number in parentheses are factors)

1(3) + 1(3) = 1 + 1 = 2

I assumed that at least one of the integers was going to be a 1.

So, I created a chart with the cubes of the first 20 integers on each side, x and y, and their sums.

This is what I came up with.

1(3) + 12(3) = 1 + 1728 = 1729

9(3) + 10(3) = 729 + 1000 = 1729

So, the number of Hardy's taxicab was 1729.

There is some interesting series here, I think. Give me another couple of hours and I think I can write a program that will figure the next number. It will be enormous, of course, and the one after that will be even bigger. I'm assuming I'm going to need a Cray-3 supercomputer shortly to figure out the next ones.

1(3) + 1(3) = 1 + 1 = 2

1(3) + 12(3) = 1 + 1728 = 1729
9(3) + 10(3) = 729 + 1000 = 1729

22:10  
Blogger Bob Merkin said...

I know what you do for a living, and am presuming from that that you have strictly abided by the Vleeptron Honor Code, no research, no Googling, no phoning Klaas in Rotterdam.

That said ...

THIS PIZZA IS YOURS!!! YOU RAWK!!! 1729 was sho' nuff the number of Hardy's Taxicab!!!

Look ... over the years I've grudgingly come to accept your Career as worthwhile and valuable and all that ... but as long as you're doing it at a university ... why don't you stroll to another part of the campus a few hours a week and take a goddam math course?

The history of Clergy who should have been tending their flocks but were conducting scientific experiments (I know you have a telescope) and messing around with mathematics (e.g. Mersenne) is a very long and rich history, and human knowledge is much the richer for it. Priestly in particular fell into trouble with his parishioners because to do his experiments with carbon dioxide, he built a direct pipe from the brewery next door to his rectory, and this caused some talk. I don't know if his parishioners finally learned that he had (among other things) invented soda pop.

Is "Grizzly Man" still playing in Beantown? I think I might have missed it out here. I could buy you your Pizza there.

00:34  
Blogger Bob Merkin said...

Oh let me be picky picky picky. You're a little rusty on the nomenclature and vocabulary.

if we don't have superscripts (if we have to write this all on the same line), we write

5 x 5 x 5 = 5^3

where 3 is the *power* or *exponent*

in 5^3 = 5 x 5 x 5 = 125 , 5 is a *factor* 3 times.

other than that, you nailed this sucker.

00:44  
Anonymous Uwe Bressem said...

Uwe Bressem has sent you a link to a weblog:

Dear Bob,

after doing my homework-which is reading your blog on a daily basis-I say: " Ron's stamp is a "Kracher"! Amazing.

Uwe

01:01  
Blogger Bob Merkin said...

Wie gehts Uwe! Danke fur Leaving A Comment! (I get depressed when nobody leaves a comment, I get Happy when people leave comments.)

Yeah, see? You're not the only Visualkunstler I know!!! Ron and I washed dishes and pots and pans in the mess hall together! (Actually not, the Navy cooked all our food for us, we were Soldaten on a Navy base.)

I really feel terrible that I exposed Ron to these Faux Postage Stamps, now he can't stop. You guys should exchange notes. Or send each other mail! THAT'S an idea!!!

He probably doesn't know *anything* about sending mail by derigibles, airplanes and rockets.

01:09  
Anonymous Jim Olson said...

Come on in to the City....Darrick and I would love to see you here..we'll even cook! We have membership at the Museum of Fine Arts, so we could spend the afternoon there...

Oh, and the next number is
87539319
= 167(3) + 436(3)
= 228(3) + 423(3)
= 255(3) + 414(3)

I love my Mac Mini!

09:32  
Blogger Bob Merkin said...

Okay listen carefully ... if you want people to think you are a Cowboy, you have to wear a Cowboy Soot. If you want people to think you are an Astronaut, you have to wear an Astronaut Soot. If you want people to think you are a Mathematician, you have to Walk The Walk and Talk The Talk.

167(3) means 167 x 3 (but in a way almost nobody would use)

You got to use the Little Carrot above the [6] key.

167^3 means 167 x 167 x 167

So you tell the World (and there are People in the World who are actually interested ... and of course a few people who would run to their pencil and paper and computer to try to prove you're wrong, so they can snicker at you):

87539319
= 167^3 + 436^3
= 228^3 + 423^3
= 255^3 + 414^3

write it this way and NOW people will think: "He must be a Mathematician!"

(with apologies to Gilbert & Sullivan:

Oh he might have been a Cowboy
A Cabdriver or a Flyboy
But in spite of all Temptations
To try other Occupations
He remains a Mathematician!
He remains a Mathematician!

That's a Big Number! But not hard to keep around the house. The Chudnovsky Brothers kept their record-breaking billion digits of pi, courtesy of Ramanujan's formula, in a hard disk. A New Yorker reporter asked them where they keep it, and one of the Brothers handed him the hard disk. "It's in there," he said.

11:20  
Blogger Bob Merkin said...

"It's a small world. But you wouldn't want to paint it."

-- Steven Wright

11:30  
Anonymous Jim Olson said...

Right. Sorry. It was early this morning when I saw that the computer had churned up the third number.

The fourth number is

6963472309248
= 2421^3 + 19083^3
= 5436^3 + 18948^3
= 10200^3 + 18072^3
= 13322^3 + 16630^3

And the fifth number is

48988659276962496
= 38787^3 + 365757^3
= 107839^3 + 362753^3
= 205292^3 + 342952^3
= 221424^3 + 336588^3
= 231518^3 + 331954^3

This is where the computer stopped. I think it reached the limit of the number of places the little calculator brain will handle.

17:37  
Blogger Bob Merkin said...

Every computer has implemented in its chip and operating system software a routine called Floating Point Arithmetic, and it's designed to handle a very healthy ballpark of arithmetic problems that would satisfy almost all users almost all the time.

But now you're leaving that comfortable everyday arithmetic world and starting to pester your computer with some very tough questions involving very big integers -- and you're going beyond the bounds of the built-in FPA routine.

Fear Not! I bring you Good Tidings!

You can program your own special integer arithmetic routines using strings rather than numbers -- treating 42376099348861 NOT as a number, but exactly the way you and your computer would manipulate the word SUPERCALLIFRAGIALISTICEXPIALIDOCIOUS. Strings can easily be far longer than the longest integer your FPA can handle. I've written programs to handle integers with thousands of digits.

Then you'll have to enroll your computer in elementary school all over again and, from scratch, teach it addition, subtraction, multiplication and division -- only this time with very long non-number strings.

I still prefer Aspirin. But maybe you're a Tylenol guy. Have fun! Break a leg! Muita merda!

Sincerity ON: I'm really glad you're enjoying Hardy's Taxicab! The first PizzaQ was pretty easy, but it's opened up one of those wonderful Number Theory windows that just floods your mind with other questions. My pal John DiP just challenged me to find ...

Hardy's Troika -- the smallest number which can be expressed as the sum of two cubes in THREE different ways! Ouch. Oooh. Ow. It hurts my brane.

18:01  
Anonymous Jim Olson said...

Yah. there seems to be a whole lot of this sort of question....

I emailed a math geek friend of mine about this and he sent me a programme that will calculate all sorts of bigger integers using string theory arithmetic.

I like this stuff. It's not algebra, just pure arithmetic.

John DiP's question is going to take some more time, but I think its basically the same thing....create a chart that summs cubes and see what the smallest number is. Obviously, one of them will contain 1^3.

18:56  
Anonymous Ron Bizer said...

Bob,
or is it Joe… a place of honor on Vleeptron. I really didn’t want to do the math, but I found the story of Hardy’s taxicab inspiring enough to create some commemorative postage. Anyway, there are a couple of facts to clear up in your posting. I was not an art slave for Big Daddy. I worked for his major competitor from Detroit — Mouse! I later got involved in the Odd Rods trading card project as a subcontractor because of my experience painting hot rod monster shirts at Mouse! Studios.
Ron
PS—What the heck is blue smoke?

12:39  
Blogger Bob Merkin said...

According to my old Army buddy Ron Bizer, when some 10-year-old Rube child was pestering the crap out of a Carney, he would tell the child, "Hey kid, go to the other end of the Midway and ask Louie to give you a Bucket of Blue Steam, then bring it back here."

According to a (formerly) young woman from Northampton, each year when the Carnival comes to town, there's a long, ancient tradition of local 14 and 15 years old girls who use young beefy horney Carney guys for "practice." They have the wonderful virtue of always leaving town in three or four days and not ever coming back: Relationship-Lite. (Once or twice each fair season, there's some sort of police Motel raid which spoils everybody's party. The Carney's name is printed in the newspaper police log, but of course the sweet young underage thing's name is never revealed.)

16:35  
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03:52  

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