Ramanujan Numbers
and
The Taxicab Problem
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.
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)”.
The graph above shows the distribution of the first 100 Ramanujan numbers (2-way pairs) in the number field. The 100th of these Ramanujan doubles occurs at: 64^3 + 164^3 = 25^3 + 167^3 = 4,673,088. Of these first 100 Ramanujan numbers, 49 are primitive as they are not multiples of smaller solutions. Multiples of all primitive solutions can be constructed by multiplying the I, J, K, L numbers above by 2, 3, 4, 5, etc.
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.
The graph above shows the magnitude of the first 100 of these Ramanujan triples. Of these one hundred 3-way solutions, 33 are primitive including all 5 of the above examples. The 100th of these “triples” is: 3806^3 + 4708^3 = 990^3 + 5412^3 = 121^3 + 5423^3 = 159,486,393,528. (Solution is not primitive.)
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. The new version of the ramanujans.c program (see below) took 30 seconds to find Taxicab(4). (3GHz Pentium 4 running Windows XP) ) An early version of the rama4.c program ( earlier than http://www.durangobill.com/Rama4.html - and even before the version at http://web.archive.org/web/20020221182745/http://www.geocities.com/durangobill/Rama4.html ) running on an old 80386 computer actually found Taxicab(4) in 1987. (Never published.)
The graph above shows where the first 100 Ramanujan Quadruples appear in the number field. Total run time for all 100 solutions was 91 minutes. (Via the most recent optimized version of the ramanujans.c program on a 3GHz. Pentium 4.) If Taxicab(5) were plotted in the above graph, it would show up at position 143.
Of these one hundred 4-way solutions, there are 31 primitive solutions. The next 300 solutions add another 34 primitive solutions.
The graph above shows the distribution of the 737 primitive 4-way solutions within the number field out to 3.981E+22. (10^22.6) The search for 4-way and higher solutions has confirmed other known Ramanujan results out through Taxicab(6), with the search continuing beyond 3.981E+22.
The number field was segmented into standard geometric width ranges such that 5 consecutive ranges (as per tick marks) result in a factor-of-10 increase in the number field. The labels on the X axis show the log(10) of the location in the number field. For example, the “17.10” label represents the number field between 1.0E+17 and 1.585E+17. We note that log(10) of 1.0E+17 equals 17.0 and log(10) of 1.585E+17 equals 17.2. The 17.10 that is seen on the X axis is the midpoint of this range.
The plotted data points for each range are histogram counts of the number of primitive 4-way solutions within each range. For example, the data point at “Number of Solutions - 5” above the 17.10 label indicates there are five primitive 4-way solutions between 1.0E+17 and 1.585E+17.
The smooth line is a least squares exponential curve fit where A, B, C are least-squares calculated constants and X is Log(10) Number-field:
Y = A*exp((X-B)*C)
The least squares curve fit implies that the number of primitive 4-way solutions expands exponentially for every 10-fold increase in the number field. For example, the number of primitive 4-way solutions between 1.0E+19 and 1.0E+20 is about 47.2 % greater than the number of solutions between 1.0E+18 and 1.0E+19. Similarly, the number of primitive 4-way solutions between 1.0E+20 and 1.0E+21 expands by another ~47.2 %. There is no proof that this exponential curve accurately depicts what can be expected at still higher ranges, but it looks like it is an exponential function. Also, the number of primitive 5-way solutions looks like it follows a similar exponential function as you progress out into the number field. (See below)
Note: The quoted “47.2 %” growth rate is a least squares calculation based on the most recent search results. As search results are expanded with new results, the least squares calculation will be updated. Changes in this calculated growth rate are likely.
Also Note: Some primitive 4-way solutions have more than 1 combination of pairs to arrive at the same number. For example, in the first 5-way solution (below), the first 4 pairs form a primitive 4-way solution. If you instead use pairs 1, 2, 3, and 5, you have another set of 4 pairs that generates the same resultant number. When this happens, the result is only counted once for the above graphical tabulation of 4-way solutions.
Alternately, any 4 of the 5 pairs in any 5-way solution can be grouped to form a 4-way solution. If at least one of these groupings is primitive, then the result is counted as a primitive 4-way solution.
For example, in the third 5-way solution below, pairs 1, 2, 3, and 4 have a Greatest Common Divisor of 5 (hence, by themselves, are not primitive) while pairs 2, 3, 4 and 5 form a primitive 4-way solution. Thus the result is counted as a primitive 4-way solution as at least one grouping is primitive.
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 first five 5-way 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
2) 463036 663908 442848 673176 410584 685904 215678 725506 77574 731514
3) 579240 666630 543145 691295 285120 776070 233775 781785 48369 788631
4) 694554 995862 664272 1009764 615876 1028856 323517 1088259 116361 1097271
5) 926072 1327816 885696 1346352 821168 1371808 431356 1451012 155148 1463028
Equal:
(With “,”) (No “,”) Exponential
1) 48,988,659,276,962,496 48988659276962496 4.899E+16
2) 391,909,274,215,699,968 391909274215699968 3.919E+17
3) 490,593,422,681,271,000 490593422681271000 4.906E+17
4) 1,322,693,800,477,987,392 1322693800477987392 1.323E+18
5) 3,135,274,193,725,599,744 3135274193725599744 3.135E+18
Solutions 1 and 3 are primitive. Solutions 2, 4, and 5 are multiples of solution 1. The numbering system corresponds to data points in the graphs (below). The solution plots in the graphs consist of primitive solutions and assorted multiples. (Multiply the I, J, K, etc. by 2, 3, 4, etc.)
The two graphs above show all the 5-way solutions out through and including Taxicab(6) at position 194. There are a total of 193 5-way solutions less than Taxicab(6) which is the last plot. Data points consist of the 19 primitive 5-way solutions less than Taxicab(6), plus multiples thereof.
The graph above shows the distribution of known primitive 5-way solutions out to 1.0E+26. The plotting method is similar to what was used for the 4-way primitive solution graph.
Totals out to 10^22.6 (3.98E+22) are complete while results out to 1.0E+26 have been calculated using known 4-way solutions. There may be additional solutions within this calculated number range. Research by the author is continuing and the graph will be updated and expanded as results are found.
The above results were found by a computer program written by the author. The source code for ramanujans.c may be viewed here. (http://www.durangobill.com/RamanujanC.html) It includes lots of documentation on how to calculate Ramanujan Numbers. The source code for the predecessor of this more recent version was rama4.c. (http://www.durangobill.com/Rama4.html) Users may use or modify either version without restriction or obligation. I would appreciate that any published results from modifications to either program include a note attributing the original algorithm to me. (The search used an optimized version of this program. While the optimized version more than doubled the execution speed, the posted version of the program gives a less cluttered picture of the algorithm.)
The ramanujans.c program will run as shown under Windows XP if compiled with the lcc-win32 “C” compiler. (http://www.cs.virginia.edu/~lcc-win32/)
On March 15, 2006 the author posted a message in the Brown University CS Atrium Yahoo Group asking for computer time on something more powerful than the author’s Pentium 4 computer. See http://groups.yahoo.com/group/CSAtrium/message/102 If there had been any responses to this message, the author would have been the first person to verify Taxicab(6).
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, and the lowest of these is shown below. Uwe Hollerbach was the first to confirm “Taxicab(6)”, and the author has since verified this earlier result. (Including the lists of 4-way and 5-way solutions less than Taxicab(6).)
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 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.)
Christian Boyer has extended the list of known primitive 6-way solutions and has published the best known candidate for Taxicab(7). http://www.christianboyer.com/taxicab/ The author is also continuing research for additional primitive 6-way solutions as well as attempting to find a still better candidate for Taxicab(7). I’ll be posting the results here in the future, but nothing is likely until sometime in 2010.
One possible way of constructing “N” way solutions is to start with “N-1” way (or lower) primitive solutions and generate all possible multiples to see if a higher order solution shows up. In addition to expanding the list of known 4-way solutions, the author is also using this method for possible solutions. If/when something of significance is found, the results will be updated on this web page.
Addition information on Ramanujan Numbers and the Taxicab Problem can be found at:
Christian Boyer’s web site: http://www.christianboyer.com/taxicab/ (Includes a photo of the real Taxicab 1729)
Also:
http://euler.free.fr/taxicab.htm
http://mathworld.wolfram.com/TaxicabNumber.html
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.)
Cabtaxi(1) through Cabtaxi(9) were previously known. The author ran a search via the Cabtaxi.c program ( http://www.durangobill.com/CabtaxiC.html ) which confirmed the results shown below.
Cabtaxi(1) = 1
= 1^3 + 0^3
Cabtaxi(2) = 91
= 3^3 + 4^3
= 6^3 - 5^3
Cabtaxi(3) = 728
= 6^3 + 8^3
= 9^3 - 1^3
= 12^3 - 10^3
Cabtaxi(4) = 2,741,256
= 108^3 + 114^3
= 140^3 - 14^3
= 168^3 - 126^3
= 207^3 - 183^3
Cabtaxi(5) = 6,017,193
= 166^3 + 113^3
= 180^3 + 57^3
= 185^3 - 68^3
= 209^3 - 146^3
= 246^3 - 207^3
Cabtaxi(6) = 1,412,774,811
= 963^3 + 804^3
= 1,134^3 - 357^3
= 1,155^3 - 504^3
= 1,246^3 - 805^3
= 2,115^3 - 2,004^3
= 4,746^3 - 4,725^3
Cabtaxi(7) = 11,302,198,488
= 1,926^3 + 1,608^3
= 1,939^3 + 1,589^3
= 2,268^3 - 714^3
= 2,310^3 - 1,008^3
= 2,492^3 - 1,610^3
= 4,230^3 - 4,008^3
= 9,492^3 - 9,450^3
Cabtaxi(8) = 137,513,849,003,496
= 22,944^3 + 50,058^3
= 36,547^3 + 44,597^3
= 36,984^3 + 44,298^3
= 52,164^3 - 16,422^3
= 53,130^3 - 23,184^3
= 57,316^3 - 37,030^3
= 97,290^3 - 92,184^3
= 218,316^3 - 217,350^3
Cabtaxi(9) = 424,910,390,480,793,000
= 645,210^3 + 538,680^3
= 649,565^3 + 532,315^3
= 752,409^3 - 101,409^3
= 759,780^3 - 239,190^3
= 773,850^3 - 337,680^3
= 834,820^3 - 539,350^3
= 1,417,050^3 - 1,342,680^3
= 3,179,820^3 - 3,165,750^3
= 5,960,010^3 - 5,956,020^3
Cabtaxi(10) has been confirmed by the author’s computer program and is equal to:
933,528,127,886,302,221,000
= 7,002,840^3 + 8,387,730^3
= 6,920,095^3 + 8,444,345^3
= 77,480,130^3 - 77,428,260^3
= 41,337,660^3 - 41,154,750^3
= 18,421,650^3 - 17,454,840^3
= 10,852,660^3 - 7,011,550^3
= 10,060,050^3 - 4,389,840^3
= 9,877,140^3 - 3,109,470^3
= 9,781,317^3 - 1,318,317^3
= 9,773,330^3 - 84,560^3
Christian Boyer had previously calculated a list of primitive 9-way solutions less than his candidate for Cabtaxi(10).
http://www.christianboyer.com/taxicab
http://www.christianboyer.com/taxicab/ListCabtaxi9_10.txt
(As displayed on the above web page)
# Ways Number
1 9 424910390480793000
2 9 825001442051661504
3 9 1153657786768695936
4 9 6123582409620312000
5 9 7468225023090417768
6 9 7545659922519832512
7 9 10933313592720956472
8 9 24326499458358849024
9 9 41359077767838467448
10 9 45307115612467444008
11 9 49308192936614146752
12 9 186525463571696587968
13 9 270266803327651272408
14 9 272257988363832744000
15 9 293071805905425386112
16 9 346083762520724574528
17 9 445079976262957683648
18 9 572219233725765415608
19 9 842751835937888190552
20 10 933528127886302221000
The program confirmed that Christian’s list of primitive 9-way solutions is in fact complete, and that his candidate for Cabtaxi(10) is in fact the lowest primitive 10-way solution.
The source code for the author’s Cabtaxi computer program is at http://www.durangobill.com/CabtaxiC.html - lots of documentation. (The Skulltrail computer ran multiple copies using a slightly different version.) The program will run as is without modification if compiled with the lcc-win32 “C” compiler. http://www.cs.virginia.edu/~lcc-win32/ )
Uwe Hollerbach also ran a search for Cabtaxi(10) at the same time that the author’s search was running. He had access to a computer cluster (much greater computing processing power) with the result that his search finished before the author’s search..
At one point, the search/confirmation for Cabtaxi(10) was tentatively proposed to be a cooperative venture with Christian Boyer, Uwe Hollerbach, and myself as co-contributors with the results jointly announced. This was during the early portion of the search when (and as documented by) E-mail conversations between Uwe Hollerbach, Christian Boyer, and the author (Bill Butler) showed that I had a significantly higher search rate than what Mr. Hollerbach was getting running multiple copies of his program on a server plus another couple of copies on an Itanium based machine.
However, after Uwe obtained access to a computer cluster (much greater computing power) to run his program, Uwe sent me an E-mail which is quoted in full below. (The initial text is a portion of the transmission protocol.)
I guess “I also have no particular reason to trust you” and “Each file will be encrypted, with a separate password, which I will not send to you (and Christian) until you've completed the corresponding portion of the number range.” rules out a joint announcement. I will however share my results with Christian.
Addition information on Ramanujan Numbers and the Taxicab Problem can be found at:
Christian Boyer’s web site http://www.christianboyer.com/taxicab/ (Includes a photo of the real Taxicab 1729)
http://euler.free.fr/taxicab.htm
http://mathworld.wolfram.com/TaxicabNumber.html
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“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)”.
The graph above shows the distribution of the first 100 Ramanujan numbers (2-way pairs) in the number field. The 100th of these Ramanujan doubles occurs at: 64^3 + 164^3 = 25^3 + 167^3 = 4,673,088. Of these first 100 Ramanujan numbers, 49 are primitive as they are not multiples of smaller solutions. Multiples of all primitive solutions can be constructed by multiplying the I, J, K, L numbers above by 2, 3, 4, 5, etc.
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.
The graph above shows the magnitude of the first 100 of these Ramanujan triples. Of these one hundred 3-way solutions, 33 are primitive including all 5 of the above examples. The 100th of these “triples” is: 3806^3 + 4708^3 = 990^3 + 5412^3 = 121^3 + 5423^3 = 159,486,393,528. (Solution is not primitive.)
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. The new version of the ramanujans.c program (see below) took 30 seconds to find Taxicab(4). (3GHz Pentium 4 running Windows XP) ) An early version of the rama4.c program ( earlier than http://www.durangobill.com/Rama4.html - and even before the version at http://web.archive.org/web/20020221182745/http://www.geocities.com/durangobill/Rama4.html ) running on an old 80386 computer actually found Taxicab(4) in 1987. (Never published.)
The graph above shows where the first 100 Ramanujan Quadruples appear in the number field. Total run time for all 100 solutions was 91 minutes. (Via the most recent optimized version of the ramanujans.c program on a 3GHz. Pentium 4.) If Taxicab(5) were plotted in the above graph, it would show up at position 143.
Of these one hundred 4-way solutions, there are 31 primitive solutions. The next 300 solutions add another 34 primitive solutions.
The graph above shows the distribution of the 737 primitive 4-way solutions within the number field out to 3.981E+22. (10^22.6) The search for 4-way and higher solutions has confirmed other known Ramanujan results out through Taxicab(6), with the search continuing beyond 3.981E+22.
The number field was segmented into standard geometric width ranges such that 5 consecutive ranges (as per tick marks) result in a factor-of-10 increase in the number field. The labels on the X axis show the log(10) of the location in the number field. For example, the “17.10” label represents the number field between 1.0E+17 and 1.585E+17. We note that log(10) of 1.0E+17 equals 17.0 and log(10) of 1.585E+17 equals 17.2. The 17.10 that is seen on the X axis is the midpoint of this range.
The plotted data points for each range are histogram counts of the number of primitive 4-way solutions within each range. For example, the data point at “Number of Solutions - 5” above the 17.10 label indicates there are five primitive 4-way solutions between 1.0E+17 and 1.585E+17.
The smooth line is a least squares exponential curve fit where A, B, C are least-squares calculated constants and X is Log(10) Number-field:
Y = A*exp((X-B)*C)
The least squares curve fit implies that the number of primitive 4-way solutions expands exponentially for every 10-fold increase in the number field. For example, the number of primitive 4-way solutions between 1.0E+19 and 1.0E+20 is about 47.2 % greater than the number of solutions between 1.0E+18 and 1.0E+19. Similarly, the number of primitive 4-way solutions between 1.0E+20 and 1.0E+21 expands by another ~47.2 %. There is no proof that this exponential curve accurately depicts what can be expected at still higher ranges, but it looks like it is an exponential function. Also, the number of primitive 5-way solutions looks like it follows a similar exponential function as you progress out into the number field. (See below)
Note: The quoted “47.2 %” growth rate is a least squares calculation based on the most recent search results. As search results are expanded with new results, the least squares calculation will be updated. Changes in this calculated growth rate are likely.
Also Note: Some primitive 4-way solutions have more than 1 combination of pairs to arrive at the same number. For example, in the first 5-way solution (below), the first 4 pairs form a primitive 4-way solution. If you instead use pairs 1, 2, 3, and 5, you have another set of 4 pairs that generates the same resultant number. When this happens, the result is only counted once for the above graphical tabulation of 4-way solutions.
Alternately, any 4 of the 5 pairs in any 5-way solution can be grouped to form a 4-way solution. If at least one of these groupings is primitive, then the result is counted as a primitive 4-way solution.
For example, in the third 5-way solution below, pairs 1, 2, 3, and 4 have a Greatest Common Divisor of 5 (hence, by themselves, are not primitive) while pairs 2, 3, 4 and 5 form a primitive 4-way solution. Thus the result is counted as a primitive 4-way solution as at least one grouping is primitive.
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 first five 5-way 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
2) 463036 663908 442848 673176 410584 685904 215678 725506 77574 731514
3) 579240 666630 543145 691295 285120 776070 233775 781785 48369 788631
4) 694554 995862 664272 1009764 615876 1028856 323517 1088259 116361 1097271
5) 926072 1327816 885696 1346352 821168 1371808 431356 1451012 155148 1463028
Equal:
(With “,”) (No “,”) Exponential
1) 48,988,659,276,962,496 48988659276962496 4.899E+16
2) 391,909,274,215,699,968 391909274215699968 3.919E+17
3) 490,593,422,681,271,000 490593422681271000 4.906E+17
4) 1,322,693,800,477,987,392 1322693800477987392 1.323E+18
5) 3,135,274,193,725,599,744 3135274193725599744 3.135E+18
Solutions 1 and 3 are primitive. Solutions 2, 4, and 5 are multiples of solution 1. The numbering system corresponds to data points in the graphs (below). The solution plots in the graphs consist of primitive solutions and assorted multiples. (Multiply the I, J, K, etc. by 2, 3, 4, etc.)
The two graphs above show all the 5-way solutions out through and including Taxicab(6) at position 194. There are a total of 193 5-way solutions less than Taxicab(6) which is the last plot. Data points consist of the 19 primitive 5-way solutions less than Taxicab(6), plus multiples thereof.
The graph above shows the distribution of known primitive 5-way solutions out to 1.0E+26. The plotting method is similar to what was used for the 4-way primitive solution graph.
Totals out to 10^22.6 (3.98E+22) are complete while results out to 1.0E+26 have been calculated using known 4-way solutions. There may be additional solutions within this calculated number range. Research by the author is continuing and the graph will be updated and expanded as results are found.
The above results were found by a computer program written by the author. The source code for ramanujans.c may be viewed here. (http://www.durangobill.com/RamanujanC.html) It includes lots of documentation on how to calculate Ramanujan Numbers. The source code for the predecessor of this more recent version was rama4.c. (http://www.durangobill.com/Rama4.html) Users may use or modify either version without restriction or obligation. I would appreciate that any published results from modifications to either program include a note attributing the original algorithm to me. (The search used an optimized version of this program. While the optimized version more than doubled the execution speed, the posted version of the program gives a less cluttered picture of the algorithm.)
The ramanujans.c program will run as shown under Windows XP if compiled with the lcc-win32 “C” compiler. (http://www.cs.virginia.edu/~lcc-win32/)
On March 15, 2006 the author posted a message in the Brown University CS Atrium Yahoo Group asking for computer time on something more powerful than the author’s Pentium 4 computer. See http://groups.yahoo.com/group/CSAtrium/message/102 If there had been any responses to this message, the author would have been the first person to verify Taxicab(6).
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, and the lowest of these is shown below. Uwe Hollerbach was the first to confirm “Taxicab(6)”, and the author has since verified this earlier result. (Including the lists of 4-way and 5-way solutions less than Taxicab(6).)
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 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.)
Christian Boyer has extended the list of known primitive 6-way solutions and has published the best known candidate for Taxicab(7). http://www.christianboyer.com/taxicab/ The author is also continuing research for additional primitive 6-way solutions as well as attempting to find a still better candidate for Taxicab(7). I’ll be posting the results here in the future, but nothing is likely until sometime in 2010.
One possible way of constructing “N” way solutions is to start with “N-1” way (or lower) primitive solutions and generate all possible multiples to see if a higher order solution shows up. In addition to expanding the list of known 4-way solutions, the author is also using this method for possible solutions. If/when something of significance is found, the results will be updated on this web page.
Addition information on Ramanujan Numbers and the Taxicab Problem can be found at:
Christian Boyer’s web site: http://www.christianboyer.com/taxicab/ (Includes a photo of the real Taxicab 1729)
Also:
http://euler.free.fr/taxicab.htm
http://mathworld.wolfram.com/TaxicabNumber.html
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.)
Cabtaxi(1) through Cabtaxi(9) were previously known. The author ran a search via the Cabtaxi.c program ( http://www.durangobill.com/CabtaxiC.html ) which confirmed the results shown below.
Cabtaxi(1) = 1
= 1^3 + 0^3
Cabtaxi(2) = 91
= 3^3 + 4^3
= 6^3 - 5^3
Cabtaxi(3) = 728
= 6^3 + 8^3
= 9^3 - 1^3
= 12^3 - 10^3
Cabtaxi(4) = 2,741,256
= 108^3 + 114^3
= 140^3 - 14^3
= 168^3 - 126^3
= 207^3 - 183^3
Cabtaxi(5) = 6,017,193
= 166^3 + 113^3
= 180^3 + 57^3
= 185^3 - 68^3
= 209^3 - 146^3
= 246^3 - 207^3
Cabtaxi(6) = 1,412,774,811
= 963^3 + 804^3
= 1,134^3 - 357^3
= 1,155^3 - 504^3
= 1,246^3 - 805^3
= 2,115^3 - 2,004^3
= 4,746^3 - 4,725^3
Cabtaxi(7) = 11,302,198,488
= 1,926^3 + 1,608^3
= 1,939^3 + 1,589^3
= 2,268^3 - 714^3
= 2,310^3 - 1,008^3
= 2,492^3 - 1,610^3
= 4,230^3 - 4,008^3
= 9,492^3 - 9,450^3
Cabtaxi(8) = 137,513,849,003,496
= 22,944^3 + 50,058^3
= 36,547^3 + 44,597^3
= 36,984^3 + 44,298^3
= 52,164^3 - 16,422^3
= 53,130^3 - 23,184^3
= 57,316^3 - 37,030^3
= 97,290^3 - 92,184^3
= 218,316^3 - 217,350^3
Cabtaxi(9) = 424,910,390,480,793,000
= 645,210^3 + 538,680^3
= 649,565^3 + 532,315^3
= 752,409^3 - 101,409^3
= 759,780^3 - 239,190^3
= 773,850^3 - 337,680^3
= 834,820^3 - 539,350^3
= 1,417,050^3 - 1,342,680^3
= 3,179,820^3 - 3,165,750^3
= 5,960,010^3 - 5,956,020^3
Cabtaxi(10) has been confirmed by the author’s computer program and is equal to:
933,528,127,886,302,221,000
= 7,002,840^3 + 8,387,730^3
= 6,920,095^3 + 8,444,345^3
= 77,480,130^3 - 77,428,260^3
= 41,337,660^3 - 41,154,750^3
= 18,421,650^3 - 17,454,840^3
= 10,852,660^3 - 7,011,550^3
= 10,060,050^3 - 4,389,840^3
= 9,877,140^3 - 3,109,470^3
= 9,781,317^3 - 1,318,317^3
= 9,773,330^3 - 84,560^3
Christian Boyer had previously calculated a list of primitive 9-way solutions less than his candidate for Cabtaxi(10).
http://www.christianboyer.com/taxicab
http://www.christianboyer.com/taxicab/ListCabtaxi9_10.txt
(As displayed on the above web page)
# Ways Number
1 9 424910390480793000
2 9 825001442051661504
3 9 1153657786768695936
4 9 6123582409620312000
5 9 7468225023090417768
6 9 7545659922519832512
7 9 10933313592720956472
8 9 24326499458358849024
9 9 41359077767838467448
10 9 45307115612467444008
11 9 49308192936614146752
12 9 186525463571696587968
13 9 270266803327651272408
14 9 272257988363832744000
15 9 293071805905425386112
16 9 346083762520724574528
17 9 445079976262957683648
18 9 572219233725765415608
19 9 842751835937888190552
20 10 933528127886302221000
The program confirmed that Christian’s list of primitive 9-way solutions is in fact complete, and that his candidate for Cabtaxi(10) is in fact the lowest primitive 10-way solution.
The source code for the author’s Cabtaxi computer program is at http://www.durangobill.com/CabtaxiC.html - lots of documentation. (The Skulltrail computer ran multiple copies using a slightly different version.) The program will run as is without modification if compiled with the lcc-win32 “C” compiler. http://www.cs.virginia.edu/~lcc-win32/ )
(Click on the above small image to see a full size image which shows)
(4 copies of the CabtaxiC program working on the Cabtaxi problem.)
The picture above shows interim Cabtaxi search results as of May 8, 2008.
(4 copies of the CabtaxiC program working on the Cabtaxi problem.)
The picture above shows interim Cabtaxi search results as of May 8, 2008.
Uwe Hollerbach also ran a search for Cabtaxi(10) at the same time that the author’s search was running. He had access to a computer cluster (much greater computing processing power) with the result that his search finished before the author’s search..
At one point, the search/confirmation for Cabtaxi(10) was tentatively proposed to be a cooperative venture with Christian Boyer, Uwe Hollerbach, and myself as co-contributors with the results jointly announced. This was during the early portion of the search when (and as documented by) E-mail conversations between Uwe Hollerbach, Christian Boyer, and the author (Bill Butler) showed that I had a significantly higher search rate than what Mr. Hollerbach was getting running multiple copies of his program on a server plus another couple of copies on an Itanium based machine.
However, after Uwe obtained access to a computer cluster (much greater computing power) to run his program, Uwe sent me an E-mail which is quoted in full below. (The initial text is a portion of the transmission protocol.)
Received: by 10.142.69.16 with SMTP id r16mr1630383wfa.268.1210286819362;
Thu, 08 May 2008 15:46:59 -0700 (PDT)
Received: by 10.142.43.19 with HTTP; Thu, 8 May 2008 15:46:59 -0700 (PDT)
Message-ID: <65d7a7e0805081546s365ec634i8bb08f8543ed31e@mail.gmail.com>
Date: Thu, 8 May 2008 15:46:59 -0700
From: "Uwe Hollerbach" <uhollerbach@gmail.com>
To: "Bill & Lisa" <lisabill@mydurango.net>,
"Christian Boyer" <cboyer@club-internet.fr>
Subject: Re: Update on Cabtaxi search (fwd)
In-Reply-To: <Pine.BSO.4.63.0805080723080.6425@anansi.hollerbach.org>
MIME-Version: 1.0
Content-Type: multipart/alternative;
boundary="----=_Part_3368_19238951.1210286819373"
References: <Pine.BSO.4.63.0805080723080.6425@anansi.hollerbach.org>
Return-Path: uhollerbach@gmail.com
X-OriginalArrivalTime: 08 May 2008 22:49:36.0143 (UTC) FILETIME=[CA2311F0:01C8B15D]
No, Bill, I'm sorry, but this doesn't work for me. Right now, you have no reason to trust me, so you may regard this just as a trick to get you to abandon your current search, and start over, so that I can get more time to finish my calculations. (After all, we're on the internet, no one knows I'm a dog.) I intend to address this, quite soon (within six to eight days), by sending you a set of a dozen or fifteen files which together contain all of the five-way or higher sums from 0 to cabtaxi(10) -- my main calculation should end before the end of next week, although I will still be verifying stuff with other calculations. Each file will be encrypted, with a separate password, which I will not send to you (and Christian) until you've completed the corresponding portion of the number range. However, in order to at least approximately prove my bona fides, once I have those files and have sent them to you, I will ask each of you to pick one of those fifteenish files (whose ranges I will have identified for you), and I will send you both the passwords for those two files. Since you will have all of the data up front, albeit encrypted, that plus your selection of two files should prove to you at least with a reasonably high probability that I have in fact finished; so past that point, you should be able to convince yourself (selves) that you can actually trust me, and that, as of six to eight days from now, I will have actually finished the calculation; after that point, as work progresses I will send you more passwords, so your trust in my calculations should be able to continually grow. Until those days have passed, I am not asking you to start over, just to keep going with your existing calculations, but collecting 5-way solutions instead of just 9-way solutions. Yes, it may slow your runs down a little bit, but over the course of a week that slowdown should amount to very little -- an hour to a day, roughly? and there is absolutely no programming involved. So if at the end of next week I have not yet sent you my data, you can then go back to the way you're calculating now, with very little loss of forward progress.
However, just as you have no particular reason to trust me, I also have no particular reason to trust you, and you are asking me to delay my forward progress for several months, and allow you to completely finish your current series of calculations (which will do me almost no good in terms of verification, but could allow you to publish your own results separately) before you re-start your calculations in a way that'll benefit me. I'm sorry, but that won't do. I'm afraid I must insist that you start saving 5-way sums immediately, and be prepared to start sending them to Christian as soon as he gets back, so that we can do actual comparisons. As I said, the additional work of saving 5-way sums should be small, and you need not do any programming work at all to change the format; I am quite prepared to write a small perl program that will convert your format into mine or Christian's, I think it would be an hour or two of work at the most.
But you need to come up with a way to demonstrate that I can & should trust you, and your current proposed schedule does not do that. If you are unable to come up with any such way, either what I wrote above or some alternative, then I'm afraid we don't have a deal. There is no need to respond immediately, you should probably think about it a little bit first; but I would like to hear from you within a week or so.
Uwe
Thu, 08 May 2008 15:46:59 -0700 (PDT)
Received: by 10.142.43.19 with HTTP; Thu, 8 May 2008 15:46:59 -0700 (PDT)
Message-ID: <65d7a7e0805081546s365ec634i8bb08f8543ed31e@mail.gmail.com>
Date: Thu, 8 May 2008 15:46:59 -0700
From: "Uwe Hollerbach" <uhollerbach@gmail.com>
To: "Bill & Lisa" <lisabill@mydurango.net>,
"Christian Boyer" <cboyer@club-internet.fr>
Subject: Re: Update on Cabtaxi search (fwd)
In-Reply-To: <Pine.BSO.4.63.0805080723080.6425@anansi.hollerbach.org>
MIME-Version: 1.0
Content-Type: multipart/alternative;
boundary="----=_Part_3368_19238951.1210286819373"
References: <Pine.BSO.4.63.0805080723080.6425@anansi.hollerbach.org>
Return-Path: uhollerbach@gmail.com
X-OriginalArrivalTime: 08 May 2008 22:49:36.0143 (UTC) FILETIME=[CA2311F0:01C8B15D]
No, Bill, I'm sorry, but this doesn't work for me. Right now, you have no reason to trust me, so you may regard this just as a trick to get you to abandon your current search, and start over, so that I can get more time to finish my calculations. (After all, we're on the internet, no one knows I'm a dog.) I intend to address this, quite soon (within six to eight days), by sending you a set of a dozen or fifteen files which together contain all of the five-way or higher sums from 0 to cabtaxi(10) -- my main calculation should end before the end of next week, although I will still be verifying stuff with other calculations. Each file will be encrypted, with a separate password, which I will not send to you (and Christian) until you've completed the corresponding portion of the number range. However, in order to at least approximately prove my bona fides, once I have those files and have sent them to you, I will ask each of you to pick one of those fifteenish files (whose ranges I will have identified for you), and I will send you both the passwords for those two files. Since you will have all of the data up front, albeit encrypted, that plus your selection of two files should prove to you at least with a reasonably high probability that I have in fact finished; so past that point, you should be able to convince yourself (selves) that you can actually trust me, and that, as of six to eight days from now, I will have actually finished the calculation; after that point, as work progresses I will send you more passwords, so your trust in my calculations should be able to continually grow. Until those days have passed, I am not asking you to start over, just to keep going with your existing calculations, but collecting 5-way solutions instead of just 9-way solutions. Yes, it may slow your runs down a little bit, but over the course of a week that slowdown should amount to very little -- an hour to a day, roughly? and there is absolutely no programming involved. So if at the end of next week I have not yet sent you my data, you can then go back to the way you're calculating now, with very little loss of forward progress.
However, just as you have no particular reason to trust me, I also have no particular reason to trust you, and you are asking me to delay my forward progress for several months, and allow you to completely finish your current series of calculations (which will do me almost no good in terms of verification, but could allow you to publish your own results separately) before you re-start your calculations in a way that'll benefit me. I'm sorry, but that won't do. I'm afraid I must insist that you start saving 5-way sums immediately, and be prepared to start sending them to Christian as soon as he gets back, so that we can do actual comparisons. As I said, the additional work of saving 5-way sums should be small, and you need not do any programming work at all to change the format; I am quite prepared to write a small perl program that will convert your format into mine or Christian's, I think it would be an hour or two of work at the most.
But you need to come up with a way to demonstrate that I can & should trust you, and your current proposed schedule does not do that. If you are unable to come up with any such way, either what I wrote above or some alternative, then I'm afraid we don't have a deal. There is no need to respond immediately, you should probably think about it a little bit first; but I would like to hear from you within a week or so.
Uwe
I guess “I also have no particular reason to trust you” and “Each file will be encrypted, with a separate password, which I will not send to you (and Christian) until you've completed the corresponding portion of the number range.” rules out a joint announcement. I will however share my results with Christian.
Addition information on Ramanujan Numbers and the Taxicab Problem can be found at:
Christian Boyer’s web site http://www.christianboyer.com/taxicab/ (Includes a photo of the real Taxicab 1729)
http://euler.free.fr/taxicab.htm
http://mathworld.wolfram.com/TaxicabNumber.html
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