The tenth cabtaxi number is 933528127886302221000

The tenth cabtaxi number is 933528127886302221000

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

    77480130^3 - 77428260^3
    41337660^3 - 41154750^3
    18421650^3 - 17454840^3
    10852660^3 - 7011550^3
    10060050^3 - 4389840^3
    9877140^3 - 3109470^3
    9781317^3 - 1318317^3
    9773330^3 - 84560^3
    8444345^3 + 6920095^3
    8387730^3 + 7002840^3

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           = -86^3 + 89^3
                    = -2^3 + 41^3
                    = 17^3 + 40^3
    439926932718    = -9955^3 + 11257^3
                    = -6625^3 + 9007^3
                    = -1801^3 + 7639^3
    3607745483160   = -27864^3 + 29334^3
                    = -13633^3 + 18313^3
                    = -2029^3 + 15349^3
    5953456282536   = -36083^3 + 37547^3
                    = -23321^3 + 26513^3
                    = 458^3 + 18124^3

and the four 5-primes solutions are

    7070014626807314            = -169859^3 + 228757^3
                                = -92399^3 + 198817^3
                                = 144379^3 + 159535^3
    104886396577678268754       = -4579915^3 + 5857309^3
                                = -2216057^3 + 4873763^3
                                = 1259051^3 + 4685887^3
    319969457707558764294       = -18513883^3 + 18819961^3
                                = -9018049^3 + 10174807^3
                                = -3145699^3 + 7054657^3
    771993128736890238906       = -22626413^3 + 23118287^3
                                = 3838075^3 + 8943911^3
                                = 6113857^3 + 8160617^3

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:

    506433677359393
    137904678696613339
    39673716067958900347
    95793117931950826933
    118371660133721558473
    624256195206357000469
    633753337814536954807
    921288078249108802505

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 = Z^3 + 1 = X^3 + Y^3. 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 = Z^3 - 1 = X^3 + Y^3. 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 = I^3 + J^3 and allowing negative I or J, I rewrote it as K = J + I,
L = J - I, then S = K*(K^2 + 3*L^2)/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
http://www.korgwal.com/ramanujan/:

9- and 10-way numbers:  http://www.korgwal.com/ramanujan/nonies.95e19.dat
8-way and higher:       http://www.korgwal.com/ramanujan/octies.95e19.dat
7-way and higher:       http://www.korgwal.com/ramanujan/hepties.95e19.dat
6-way and higher:       http://www.korgwal.com/ramanujan/hexies.95e19.dat
5-way and higher:       http://www.korgwal.com/ramanujan/quints.95e19.dat
4-way and higher:       http://www.korgwal.com/ramanujan/quads.95e19.dat

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:

http://www.korgwal.com/ramanujan/all_by_42.dat

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 = Ai^3 + Bi^3 for all i; the only difference from the earlier
calculations is that now the Ai can be negative.

Here are copies of the programs:

http://www.korgwal.com/ramanujan/arith4nt.c     the main hash-bucket
scanner (this uses GMP to format the ramanumbers for output, but is
otherwise pretty self-contained).

http://www.korgwal.com/ramanujan/verify1.hs: check that sums of cubes
of given components add up to ramanumber

http://www.korgwal.com/ramanujan/verify2n.hs: independently generate
components for a given ramanumber candidate

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                  <uhollerbach@...>
apprentice arithmancer

May 14, 2008