Ramanujan number

The number 1729 is indeed interesting. It is the smallest number that can be expressed as a sum of two cubes not once but twice.
1729=10^3+963
=12^3+1^3
It was discovered by Ramanujan who did extensive self-trained research on Mathematics.

This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. It was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02, as well as the robot character Bender's serial number, as portrayed in a Christmas card in the episode Xmas Story.


I read an article online that reflected a wee bit of the genius of this man. The link is: http://www.durangobill.com/Ramanujan.html. I am able to comprehend a bit of the pattern. However, I believe thay since there is a program in C to generate Ramanujan numbers, the algorithm for the same can be understood with a bit of hard work. With my little comprehension, I am trying to give the key features of this. Here it goes:


Durango Bill’s
Ramanujan Numbers
and
The Taxicab Problem


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If you mention the number “1729” or the phrase “Taxicab Problem” to any mathematician, it will immediately bring up the subject of the self-taught Indian mathematical genius Srinivasa Ramanujan. When Ramanujan was dying of tuberculosis in a hospital, G. H. Hardy would frequently visit him. It was on one of these visits that the following occurred according to C. P. Snow.


“Hardy used to visit him, as he lay dying in hospital at Putney. It was on one of those visits that there happened the incident of the taxicab number. Hardy had gone out to Putney by taxi, as usual his chosen method of conveyance. He went into the room where Ramanujan was lying. Hardy, always inept about introducing a conversation, said, probably without a greeting, and certainly as his first remark: ‘I thought the number of my taxicab was 1729. It seemed to me rather a dull number.’ To which Ramanujan replied: ‘No, Hardy! No, Hardy! It is a very interesting number. It is the smallest number expressible as the sum of two cubes in two different ways.’”


Since then, integer solutions to:

I^3 + J^3 = K^3 + L^3

have been called “Ramanujan Numbers”.

The first five of these are:
Ramanujan Number
I J K L (No “,” With “,”)
----------------------------------------------
1 12 9 10 1729 1,729
2 16 9 15 4104 4,104
2 24 18 20 13832 13,832 (This is a multiple of Solution 1)
10 27 19 24 20683 20,683
4 32 18 30 32832 32,832 (This is a multiple of Solution 2)

The lowest solution to this “2-way” problem is also referred to as “Taxicab(2)”.



Ramanujan Triples

Next, we might ask if there are any triple pair solutions to I^3 + J^3 = K^3 + L^3 = M^3 + N^3 where all the numbers are integers. Again, there are an infinite number of solutions. The first 5 solutions are:

Ramanujan Triple
I J K L M N (No “,” With “,”)
-----------------------------------------------------------------
228 423 167 436 255 414 87539319 87,539,319
11 493 90 492 346 428 119824488 119,824,488
111 522 408 423 359 460 143604279 143,604,279
70 560 198 552 315 525 175959000 175,959,000
339 661 300 670 510 580 327763000 327,763,000

Solutions involving 3 pairs are also called 3-way solutions. The lowest solution to any “N-Way” problem is also called a “Taxicab Number”. Thus “Taxicab(3)” is 87539319.



Ramanujan Quadruples

The sequence can be extended through Ramanujan Quadruples. (There are 4 ways that the sum of two cubes can share a common sum.) The first five quadruple pairs (I^3 + J^3 = K^3 + L^3 = M^3 + N^3 = O^3 + P^3) are:

Ramanujan
I J K L M N O P Quadruple
-----------------------------------------------------------------------
13322 16630 10200 18072 5436 18948 2421 19083 6,963,472,309,248
12939 21869 10362 22580 7068 23066 4275 23237 12,625,136,269,928
17176 25232 11772 26916 8664 27360 1539 27645 21,131,226,514,944
21930 24940 14577 28423 12900 28810 4170 29620 26,059,452,841,000
26644 33260 20400 36144 10872 37896 4842 38166 55,707,778,473,984 (A multiple)

Taxicab(4) is thus 6963472309248.

Ramanujan Quintuples


If a number can be formed by the sum of 2 cubes in 5 different ways (5-way solution) it becomes a Ramanujan Quintuple. The 16 lowest primitive solutions are shown in the table below. The lowest is of course “Taxicab(5)” which has been found/verified by several sources. The ramanujans.c program took 3 hrs. 15 min. for Taxicab(5). (The current optimized version cuts this to less than 2 hours.)

(I^3 + J^3 = K^3 + L^3 = M^3 + N^3 = O^3 + P^3 = Q^3 + R^3)

I J K L M N O P Q R
-----------------------------------------------------------------------------------------------
1) 231518 331954 221424 336588 205292 342952 107839 362753 38787 365757
3) 579240 666630 543145 691295 285120 776070 233775 781785 48369 788631
8) 1462050 1478238 1150792 1690544 788724 1803372 580488 1833120 103113 1852215
13) 1872184 2750288 1283148 2933844 944376 2982240 265392 3012792 167751 3013305
16) 2808000 2953080 2384460 3250260 2025400 3408080 1041204 3602796 262665 3631095
19) 2273733 3527139 1941984 3642078 1654136 3711070 1329636 3762990 653022 3811152
22) 2615985 3692391 1839516 3958290 1164002 4054792 640500 4081266 120069 4086483
34) 4972160 5227585 3884265 5917170 2595033 6285342 2416890 6313545 1006145 6421240
35) 4542802 5670830 3478200 6162552 1853676 6461268 825561 6507303 581384 6510184
38) 3811712 6608416 3126048 6792768 2658867 6876621 1509320 6983224 1084848 6997968
39) 5486400 5769864 4658868 6350508 3957320 6658864 3325590 6843114 513207 7094601
46) 5966610 6293820 5348655 6758505 3469365 7488675 2641964 7624786 225810 7729020
50) 5708052 7282590 5384475 7465677 4989264 7651854 4016670 7976052 3179918 8143576
53) 5167575 8016225 4413600 8277450 4112052 8356698 3759400 8434250 3021900 8552250
57) 6461170 8065550 4947000 8764920 2636460 9189780 1405246 9250754 1174185 9255255
65) 8387550 8480418 6601912 9698384 3330168 10516320 935856 10624056 591543 10625865

Of interest is the increasing sparseness of numbers that can be formed by the sum of two cubes. At 1.0E+20, only one number in 16,000,000 is the sum of two cubes. The sparseness slowly gets worse with increasing number size. A Poisson Distribution calculation based on this “density” indicates a random number near 1.0E+20 should have only a 7.9E-39 probability of forming a 5-way solution. If numbers that can be formed by the sum of two cubes were distributed randomly, there probably wouldn’t be any 5-way or greater solutions. (This would be particularly true for primitive solutions beyond the first one or two.) Given that sporadic 5-way solutions exist, one can conclude that the distribution is not entirely random. (This extends beyond the known modulo 9 relationship.)

Ramanujan Sextuples


The process of “N-way” solutions can be extended to numbers that can be formed by the sum of 2 cubes in 6 different ways. There are several known solutions, but the search run by Uwe Hollerbach confirms that “Taxicab(6)” is the result shown below.

Taxicab(6) = 24153319581254312065344
= 28906206^3 + 582162^3
= 28894803^3 + 3064173^3
= 28657487^3 + 8519281^3
= 27093208^3 + 16218068^3
= 26590452^3 + 17492496^3
= 26224366^3 + 18289922^3


It is interesting to note that this candidate for Taxicab(6) is 79 times Taxicab(5). If you multiply the I, J, K, etc. components of Taxicab(5) by 79, you will get the last 5 pairs of Taxicab(6). (The actual resulting number is 79^3 times larger.)

One possible way of constructing “N” way solutions is to start with “N-1” way primitive solutions and generate/try all possible multiples to see if anything interesting happens. The author tried multiplying all the above primitive 5-way solutions by all integers such that the result was less than Taxicab(6). There were no new smaller 6-way solutions. (Note: All integer multiples have to be used for these trials and not just multiples using prime numbers. For example, the 5th primitive 5-way solution, “16)” above, is 65 times a primitive 4-way solution, and “65” is not a prime.)

Cabtaxi Numbers

While “Taxicab(N)” is defined as the lowest number that can be formed by the sum of two cubes in “N” different ways, Cabtaxi(N) is defined as the lowest number that can be formed by the sum and/or difference of two cubes in “N” different ways. (See http://en.wikipedia.org/wiki/Cabtaxi_number for more information.)