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.
(All results shown below include a computer search through 1.406E+21)
Update March 23, 2008: The graph above shows the distribution of primitive 4-way solutions within the number field out to 1.406E+21. There are 398 primitive 4-way solutions less than 1.406E+21. (The graph includes partial results in the last column.)
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 58.5 % 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 ~58.5 %. 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.
(Results from Uwe Hollerbach’s search out to Taxicab(6) (see below) add another 285 4-way solutions which modifies the exponential growth rate to 51.5 % per 10-fold increase in the number field. I will update the chart when I can verify these results with my own data. It looks like there are 683 primitive 4-way solutions out to and including Taxicab(6).)
Note: The quoted “58.5 %” 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. Minor 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 second 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 has at least one grouping that 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 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
Three other primitive solutions are known less than 2.4E+22
9258790 11557850 7965288 12236712 7089000 12560040 3778020 13168860 1682595 13262685
15076800 15855768 12802716 17451396 10874840 18298768 4795032 19401504 1410309 19496187
13696103 19489921 12579342 19984980 11238804 20450574 3349092 21497550 1832949 21520179
(With “,”) (No “,”) Exponential
1) 48,988,659,276,962,496 48988659276962496 4.899E+16
3) 490,593,422,681,271,000 490593422681271000 4.906E+17
8) 6,355,491,080,314,102,272 6355491080314102272 6.355E+18
13) 27,365,551,142,421,413,376 27365551142421413376 2.737E+19
16) 47,893,568,195,858,112,000 47893568195858112000 4.789E+19
19) 55,634,997,032,869,710,456 55634997032869710456 5.563E+19
22) 68,243,313,527,087,529,096 68243313527087529096 6.824E+19
34) 265,781,191,139,199,122,625 265781191139199122625 2.658E+20
35) 276,114,357,544,758,340,608 276114357544758340608 2.761E+20
38) 343,978,135,086,713,831,424 343978135086713831424 3.440E+20
39) 357,230,299,141,507,244,544 357230299141507244544 3.572E+20
46) 461,725,779,831,883,749,000 461725779831883749000 4.617E+20
50) 572,219,233,725,765,415,608 572219233725765415608 5.722E+20
53) 653,115,573,732,974,625,000 653115573732974625000 6.531E+20
57) 794,421,645,362,287,488,000 794421645362287488000 7.944E+20
65) 1,199,962,860,219,870,469,632 1199962860219870469632 1.200E+21
Three other primitive solutions are known less than 2.4E+22
2,337,654,192,461,288,064,000 2337654192461288064000 2.338E+21
7,413,331,235,096,863,544,832 7413331235096863544832 7.413E+21
9,972,542,662,841,658,461,688 9972542662841658461688 9.973E+21
Note: A computer search by Uwe Hollerbach (See March 25. 2008 update below) indicates that the list given here is complete and no further primitive 5-way solutions exist that are less than Taxicab(6). I will continue my plans for the exhaustive search out to Taxicab(6); but unless something unexpected shows up, Uwe’s results of no additional 5-way solutions should be assumed to be accurate. (Earlier I had assumed that at least 15 other 5-way solutions were “yet to be found”.)
The numbering system corresponds to data points in the graphs (below). The solution plots in the graphs consist of these primitive solutions and multiples of them. (Multiply the I, J, K, etc. by 2, 3, 4, etc.)
Primitives out to 1.4E+21 have been verified by an exhaustive computer search. Larger primitive solutions were found via calculations using primitive 4-way solutions. (See below)
The two graphs above show all the known 5-way solutions with Taxicab(6) at position 194. There are a total of 193 known 5-way solutions less than this candidate for Taxicab(6) which is the last plot. Data points consist of the 19 known primitive solutions plus multiples thereof.
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.)
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/)
Update Mar. 3, 2008
I’ve ordered a Mach V Skulltrail computer system (twin QX9775 3.2GHz quad core processors - 8 GB DDR2-800 RAM) from Falcon Northwest Computer Systems. http://www.falcon-nw.com/ (Machine will be configured for number crunching - not game playing. For example, see http://www.tomshardware.com/2008/02/08/intel_skulltrail_part_3/page16.html for expected benchmarks/performance for this usage.) It will be mid April before I receive the system and get everything set up. It should confirm what Taxicab(6) is some 2 to 3 months of search time after that. (Intend to use 7 cores running 24/7.) (I will have to suspend the search part way through as I will be away on a vacation trip, but the total search out to Taxicab(6) should be complete by mid August 2008 - and maybe a couple of weeks before then.)
Update March 25, 2008
On March 9, 2008 Uwe Hollerbach posted the results of his computer runs that verify that the candidate for Taxicab(6) (shown below) is in fact Taxicab(6).
http://www.nabble.com/The-sixth-taxicab-number-is-24153319581254312065344-to15947247.html
http://www.korgwal.com/ramanujan/
Also, he did not find any additional 5-way solutions beyond those shown above. Thus the expectation that I gave in earlier versions of this web page that more 5-way solutions would be found is in reality not going to prove true. I will run the search anyway just to confirm Uwe’s results. (In any endeavor, it’s always a strong confirmation if two independent sources arrive at the same result.)
On March 15, 2006 (2 years earlier) 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 confirmed Taxicab(6) 2 years before Uwe’s results. Also, please note the date (within the URL) from “The Wayback Machine”. http://web.archive.org/web/20060315212022/http://www.durangobill.com/RamanujanC.html
Update April 9, 2008
The new Falcon Northwest Skulltrail Mach V computer has arrived. I’ve written (and tested) a Cabtaxi program on my old Pentium 4 which I’ll port to the Skulltrail when it’s fully up and running in about a week. Time trials on the Pentium 4 imply that the Skulltrail will be able to search out to a known 10-way Cabtaxi solution at 9.335 E+20 in about 6 months. This search for the lowest 10-way solution will have priority over the previously mentioned Taxicab(6) search as Cabtaxi(10) is not known at present. Both the “I” and “J” in the Cabtaxi “I^3 - J^3” pairs can extend well above 2^32 which leads to “interesting” programming. (Hint: Think of efficiently calculating the I^3 - J^3 differences when the difference between I and J is small.)
The tested search rate for the Cabtaxi program using a 3 GHz (Task Manager shows the program gets just half of that.) Pentium 4 finds Cabtaxi(9) in slightly over 2 days. The Skulltrail will use 7 dedicated cores, each at 3.2 GHz plus a significant improvement in search efficiency available from more RAM to produce an estimated 20 times (possibly more) faster search rate. See the bottom portion of this page for more on Cabtaxi Numbers.
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.)
Nearly 400 primitive 4-way solutions have been found by a brute force computer search. These primitive 4-way solutions were the input to another computer program that systematically tried all multiples such that the result was <= “candidate Taxicab(6)”. Most of the known primitive 5-way solutions were initially found via these calculations. (These were reconfirmed by the brute force search.) These results also generated Taxicab(6). No combinations were found for a smaller 6-way solution. The technique is very efficient when an N-way solution is a multiple of an (N-1)-way solution, but doesn’t work at all for primitive N-way solutions that are not simple multiples. Unfortunately, the 5-way solutions that were found via this methodology showed up early in the search. Calculations using subsequent primitive 4-way solutions regenerated these earlier results, but did not find any new 5-way solutions.
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) are known and are shown below. I expect to be able to verify and display Cabtaxi(10) no later than late July, 2008.
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) will be announced here about late July, 2008
Christian Boyer has calculated a list that contains a significant portion (probably most) of the primitive 9-way Cabtaxi solutions that are <= the best known upper bound for Cabtaxi(10).
http://cboyer.club.fr/Taxicab.htm
http://cboyer.club.fr/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 author’s computer program has been verifying that this is in fact a pretty good list. If/when new primitive 9-way solutions are found, they will be posted here, and the search out to 9.33E20 should be complete by the end of July. ( Source code at http://www.durangobill.com/CabtaxiC.html - lots of documentation. The program will run as is without modification if compiled with the lcc-win32 “C” compiler. http://www.cs.virginia.edu/~lcc-win32/ )
The picture above shows interim Cabtaxi search results as of May 8, 2008. I will eventually be able to run 7 or 8 copies simultaneously on the Skulltrail after I am able to upgrade from Windows Vista Home Premium to Windows Vista Ultimate. (Long story.) I will be away on an extended vacation trip from late May to late June and will have to shut the search down for a month, but I still expect to be able to confirm Cabtaxi(10) by late July, 2008.
Uwe Hollerbach has also been running a search for Cabtaxi(10). He has access to a computer cluster (much greater computing processing power), so there is a distinct possibility that he may be able to confirm Cabtaxi(10) before I post the results here.
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://cboyer.club.fr:80/Taxicab.htm (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.
(All results shown below include a computer search through 1.406E+21)
Update March 23, 2008: The graph above shows the distribution of primitive 4-way solutions within the number field out to 1.406E+21. There are 398 primitive 4-way solutions less than 1.406E+21. (The graph includes partial results in the last column.)
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 58.5 % 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 ~58.5 %. 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.
(Results from Uwe Hollerbach’s search out to Taxicab(6) (see below) add another 285 4-way solutions which modifies the exponential growth rate to 51.5 % per 10-fold increase in the number field. I will update the chart when I can verify these results with my own data. It looks like there are 683 primitive 4-way solutions out to and including Taxicab(6).)
Note: The quoted “58.5 %” 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. Minor 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 second 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 has at least one grouping that 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 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
Three other primitive solutions are known less than 2.4E+22
9258790 11557850 7965288 12236712 7089000 12560040 3778020 13168860 1682595 13262685
15076800 15855768 12802716 17451396 10874840 18298768 4795032 19401504 1410309 19496187
13696103 19489921 12579342 19984980 11238804 20450574 3349092 21497550 1832949 21520179
(With “,”) (No “,”) Exponential
1) 48,988,659,276,962,496 48988659276962496 4.899E+16
3) 490,593,422,681,271,000 490593422681271000 4.906E+17
8) 6,355,491,080,314,102,272 6355491080314102272 6.355E+18
13) 27,365,551,142,421,413,376 27365551142421413376 2.737E+19
16) 47,893,568,195,858,112,000 47893568195858112000 4.789E+19
19) 55,634,997,032,869,710,456 55634997032869710456 5.563E+19
22) 68,243,313,527,087,529,096 68243313527087529096 6.824E+19
34) 265,781,191,139,199,122,625 265781191139199122625 2.658E+20
35) 276,114,357,544,758,340,608 276114357544758340608 2.761E+20
38) 343,978,135,086,713,831,424 343978135086713831424 3.440E+20
39) 357,230,299,141,507,244,544 357230299141507244544 3.572E+20
46) 461,725,779,831,883,749,000 461725779831883749000 4.617E+20
50) 572,219,233,725,765,415,608 572219233725765415608 5.722E+20
53) 653,115,573,732,974,625,000 653115573732974625000 6.531E+20
57) 794,421,645,362,287,488,000 794421645362287488000 7.944E+20
65) 1,199,962,860,219,870,469,632 1199962860219870469632 1.200E+21
Three other primitive solutions are known less than 2.4E+22
2,337,654,192,461,288,064,000 2337654192461288064000 2.338E+21
7,413,331,235,096,863,544,832 7413331235096863544832 7.413E+21
9,972,542,662,841,658,461,688 9972542662841658461688 9.973E+21
Note: A computer search by Uwe Hollerbach (See March 25. 2008 update below) indicates that the list given here is complete and no further primitive 5-way solutions exist that are less than Taxicab(6). I will continue my plans for the exhaustive search out to Taxicab(6); but unless something unexpected shows up, Uwe’s results of no additional 5-way solutions should be assumed to be accurate. (Earlier I had assumed that at least 15 other 5-way solutions were “yet to be found”.)
The numbering system corresponds to data points in the graphs (below). The solution plots in the graphs consist of these primitive solutions and multiples of them. (Multiply the I, J, K, etc. by 2, 3, 4, etc.)
Primitives out to 1.4E+21 have been verified by an exhaustive computer search. Larger primitive solutions were found via calculations using primitive 4-way solutions. (See below)
The two graphs above show all the known 5-way solutions with Taxicab(6) at position 194. There are a total of 193 known 5-way solutions less than this candidate for Taxicab(6) which is the last plot. Data points consist of the 19 known primitive solutions plus multiples thereof.
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.)
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/)
Update Mar. 3, 2008
I’ve ordered a Mach V Skulltrail computer system (twin QX9775 3.2GHz quad core processors - 8 GB DDR2-800 RAM) from Falcon Northwest Computer Systems. http://www.falcon-nw.com/ (Machine will be configured for number crunching - not game playing. For example, see http://www.tomshardware.com/2008/02/08/intel_skulltrail_part_3/page16.html for expected benchmarks/performance for this usage.) It will be mid April before I receive the system and get everything set up. It should confirm what Taxicab(6) is some 2 to 3 months of search time after that. (Intend to use 7 cores running 24/7.) (I will have to suspend the search part way through as I will be away on a vacation trip, but the total search out to Taxicab(6) should be complete by mid August 2008 - and maybe a couple of weeks before then.)
Update March 25, 2008
On March 9, 2008 Uwe Hollerbach posted the results of his computer runs that verify that the candidate for Taxicab(6) (shown below) is in fact Taxicab(6).
http://www.nabble.com/The-sixth-taxicab-number-is-24153319581254312065344-to15947247.html
http://www.korgwal.com/ramanujan/
Also, he did not find any additional 5-way solutions beyond those shown above. Thus the expectation that I gave in earlier versions of this web page that more 5-way solutions would be found is in reality not going to prove true. I will run the search anyway just to confirm Uwe’s results. (In any endeavor, it’s always a strong confirmation if two independent sources arrive at the same result.)
On March 15, 2006 (2 years earlier) 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 confirmed Taxicab(6) 2 years before Uwe’s results. Also, please note the date (within the URL) from “The Wayback Machine”. http://web.archive.org/web/20060315212022/http://www.durangobill.com/RamanujanC.html
Update April 9, 2008
The new Falcon Northwest Skulltrail Mach V computer has arrived. I’ve written (and tested) a Cabtaxi program on my old Pentium 4 which I’ll port to the Skulltrail when it’s fully up and running in about a week. Time trials on the Pentium 4 imply that the Skulltrail will be able to search out to a known 10-way Cabtaxi solution at 9.335 E+20 in about 6 months. This search for the lowest 10-way solution will have priority over the previously mentioned Taxicab(6) search as Cabtaxi(10) is not known at present. Both the “I” and “J” in the Cabtaxi “I^3 - J^3” pairs can extend well above 2^32 which leads to “interesting” programming. (Hint: Think of efficiently calculating the I^3 - J^3 differences when the difference between I and J is small.)
The tested search rate for the Cabtaxi program using a 3 GHz (Task Manager shows the program gets just half of that.) Pentium 4 finds Cabtaxi(9) in slightly over 2 days. The Skulltrail will use 7 dedicated cores, each at 3.2 GHz plus a significant improvement in search efficiency available from more RAM to produce an estimated 20 times (possibly more) faster search rate. See the bottom portion of this page for more on Cabtaxi Numbers.
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.)
Nearly 400 primitive 4-way solutions have been found by a brute force computer search. These primitive 4-way solutions were the input to another computer program that systematically tried all multiples such that the result was <= “candidate Taxicab(6)”. Most of the known primitive 5-way solutions were initially found via these calculations. (These were reconfirmed by the brute force search.) These results also generated Taxicab(6). No combinations were found for a smaller 6-way solution. The technique is very efficient when an N-way solution is a multiple of an (N-1)-way solution, but doesn’t work at all for primitive N-way solutions that are not simple multiples. Unfortunately, the 5-way solutions that were found via this methodology showed up early in the search. Calculations using subsequent primitive 4-way solutions regenerated these earlier results, but did not find any new 5-way solutions.
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) are known and are shown below. I expect to be able to verify and display Cabtaxi(10) no later than late July, 2008.
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) will be announced here about late July, 2008
Christian Boyer has calculated a list that contains a significant portion (probably most) of the primitive 9-way Cabtaxi solutions that are <= the best known upper bound for Cabtaxi(10).
http://cboyer.club.fr/Taxicab.htm
http://cboyer.club.fr/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 author’s computer program has been verifying that this is in fact a pretty good list. If/when new primitive 9-way solutions are found, they will be posted here, and the search out to 9.33E20 should be complete by the end of July. ( Source code at http://www.durangobill.com/CabtaxiC.html - lots of documentation. 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.)
(As of 10:45 PM MDT May 9, 2008, all search segments have progressed to the number field above 4.0E20)
(All search segments should be above 5.0E20 by early afternoon May 17, 2008)
(4 copies of the CabtaxiC program working on the Cabtaxi problem.)
(As of 10:45 PM MDT May 9, 2008, all search segments have progressed to the number field above 4.0E20)
(All search segments should be above 5.0E20 by early afternoon May 17, 2008)
The picture above shows interim Cabtaxi search results as of May 8, 2008. I will eventually be able to run 7 or 8 copies simultaneously on the Skulltrail after I am able to upgrade from Windows Vista Home Premium to Windows Vista Ultimate. (Long story.) I will be away on an extended vacation trip from late May to late June and will have to shut the search down for a month, but I still expect to be able to confirm Cabtaxi(10) by late July, 2008.
Uwe Hollerbach has also been running a search for Cabtaxi(10). He has access to a computer cluster (much greater computing processing power), so there is a distinct possibility that he may be able to confirm Cabtaxi(10) before I post the results here.
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)
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Date: Thu, 8 May 2008 15:46:59 -0700
From: "Uwe Hollerbach" <uhollerbach@gmail.com>
To: "Bill & Lisa" <lisabill@mydurango.net>,
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Subject: Re: Update on Cabtaxi search (fwd)
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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://cboyer.club.fr:80/Taxicab.htm (Includes a photo of the real Taxicab 1729)
http://euler.free.fr/taxicab.htm
http://mathworld.wolfram.com/TaxicabNumber.html
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