May 14, 2008
Dear number theorists,
I am pleased to announce that I have verified that the tenth cabtaxi
number is 933528127886302221000: it is the smallest
positive number that can be expressed in ten different ways as the sum
of two (not necessarily positive) cubes. This result is subject to the
usual caveats: either I or my electronic minions could have screwed
up. I have attempted to catch any such screwups, more on that below,
but since I am only carbon, oxygen, etc., augmented with a few little
bits of silicon, it is still possible that undetected errors remain.
What have I done?
I have scanned the entire range of numbers from 1 to 9.5e20. In performing this scan, I kept all primitive three-way or higher Ramanujan numbers with both positive and negative components. In order to attempt to catch screwups by either my computers or myself, I verified some of the results of the scan with different programs than I had used for the main scan.
The main findings:
933528127886302221000, first reported by Christian Boyer in 2006, with the ten different representations
is indeed Cabtaxi(10).
I found 19 other 9-way solutions and 114 other 8-way solutions. These were all already known to Christian, who was kind enough to provide me with a list of them (out of which 10 8-way solutions had been obscured). I also found 617 7-way solutions in the range 1 to 9.5e20, of which Cabtaxi(10) is #614, 2901 6-way solutions, 14186 5-way solutions, 88968 4-way solutions, and 10597216 3-way solutions. (Each of these counts includes the higher-way solutions.)
In addition, at Christian's direction, I looked for solutions with large numbers of primes among their components: I found 54 solutions with 4 primes, of which the smallest had already been known to Christian, and 4 solutions with 5 primes, which had not previously been known. Intriguingly, the 4-primes solutions seem to occur somewhat regularly, leading us to conjecture that there may be infinitely many of them, and the 5-primes solutions seem to be occurring in a similar pattern as the 4-primes solutions: the first one is very much smaller than the rest, which then grow somewhat more sedately: the first few (3-way) 4-primes solutions are
| 68913 | = -863 + 893 |
| = -23 + 413 | |
| = 173 + 403 | |
| 439926932718 | = -99553 + 112573 |
| = -66253 + 90073 | |
| = -18013 + 76393 | |
| 3607745483160 | = -278643 + 293343 |
| = -136333 + 183133 | |
| = -20293 + 153493 | |
| 5953456282536 | = -360833 + 375473 |
| = -233213 + 265133 | |
| = 4583 + 181243 |
and the four 5-primes solutions are
| 7070014626807314 | = -1698593 + 2287573 |
| = -923993 + 1988173 | |
| = 1443793 + 1595353 | |
| 104886396577678268754 | = -45799153 + 58573093 |
| = -22160573 + 48737633 | |
| = 12590513 + 46858873 | |
| 319969457707558764294 | = -185138833 + 188199613 |
| = -90180493 + 101748073 | |
| = -31456993 + 70546573 | |
| 771993128736890238906 | = -226264133 + 231182873 |
| = 38380753 + 89439113 | |
| = 61138573 + 81606173 |
A fifth 5-primes solution, much larger than these, has recently been found by Jaroslaw Wroblewski; this might tend to support the idea that there could be an infinite number of them.
We hoped that there might be a 6-primes solution, but I did not find any; based on the above numbers, it seems plausible that there might be one, but it would probably be much much larger than any numbers I found.
I also looked for cubefree solutions (only among the 4-way-and-higher solutions) and found about 800 4-way solutions, and the following 8 5-way solutions:
There were no cubefree 6-way solutions; the smallest such is likely much larger than the range I examined.
In my previous search for Taxicab(6), I had also collected "Fermat near-misses": numbers S = Z3 + 1 = X3 + Y3. Unfortunately, this time around, since the 2-way solutions were too numerous to keep, I was not able to find the other half of the "Fermat near-misses", those numbers expressible as S = Z3 - 1 = X3 + Y3. Someone else will have to do that...
How did it all get done?
The main scan was done using a modification of the program I used for Taxicab(6) some months ago. Instead of retaining the expression as S = I3 + J3 and allowing negative I or J, I rewrote it as K = J + I, L = J - I, then S = K*(K2 + 3*L2)/4. This has the dual advantage that all quantities are again unsigned, and only one of them, L, gets larger than 32 bits (for the range I was examining).
I also tried the heap-based approach I had used in the Taxicab(6) verification; that was not as successful this time. I found that in order to get all the numbers included, the heap has to grow very quickly: it basically grows like L, rather than K, and L grows to about 3.6e10 in this calculation. This is more or less true even for heaps mod N, so it was not possible to use this as a separate check on the main calculations. I did implement the heap-based program and used it as a check on the main program for small sums, up to about 1e16; there were no discrepancies there.
In addition, I made a couple of other checks: first, I extracted all 3-way-or-higher sums including only positive components (some of these were sub-sets of higher-way sums with negative components) and compared these against the list of triples-or-higher from the earlier Taxicab(6) calculation; these also all matched.
Analogous to the final checks I did for the Taxicab(6) calculation, I also wrote two small Haskell programs to, first, verify that each recorded number is expressed correctly as the sum of the cubes of the two recorded components, for all pairs of components, and second, an independent re-generation of all components for those numbers which I checked: for both of these programs, I checked only the ~90K 4-way-or-higher sums, not the many more 3-way sums. Again, I found no differences.
I have placed copies of several data files onto my website at www.korgwal.com/ramanujan/:
The 4-way file is about 7.8 MB, the 5-way file about 1.5 MB; the others are all fairly small. In addition, I extracted every 42nd value from the full 3-way-or-higher dataset. This last file is about 18 MB large.
If you want to check my calculations (and I urge you to do so), I would suggest grabbing either the 4-way-and-higher numbers or the all-by-42 numbers and calculating some ranges. If you find any 4-way numbers that are not in my list, or you find more or less than 41 values between any two adjacent ones from the all-by-42 list, then either you or I will have screwed up... if you determine that it's me, then I would be very interested to hear from you.
In all of the above data files, the format is the same as for the earlier Taxicab(6) calculation:
S A1 B1 A2 B2 ...
where S = Ai3 + Bi3 for all i;
the only difference from the earlier calculations is that now the Ai
can be negative.
Here are copies of the programs:
The full dataset from this calculation is a little too large to put onto my website, about 750 MB; if you are interested, please contact me and we'll work out a way to get it to you.
A portion of these calculations was done using some of the idle cycles of an old Itanium server belonging to my current employer, Brion Technologies, an ASML company; in addition, I used some idle time on a modest-sized cluster, also located at and belonging to Brion. I thank them for these idle cycles. Other participating computers included a Sun server with a couple of Opteron CPUs, a couple of Pentium PCs, an old iMac G3, and a Core 2 Quad PC (the only modern computer in the gang!)
I also thank Christian Boyer for interesting and useful discussions.
Uwe Hollerbach
apprentice arithmancer
May 14, 2008