From: David desJardins <desj@math.berkeley.edu>
Subject: Re: 2, 1729, ?
Date: 01 Jan 2001 19:15:01 -0800
Newsgroups: sci.math.research
Summary: Smallest numbers representables as sums of 2 cubes in  k  ways.

Edwin Clark <eclark@math.usf.edu> writes:
> Clearly R(0) = 2. From the Hardy-Ramanujan story we know
> R(1) = 1729.

Typing 2,1729 at http://www.research.att.com/~njas/sequences/ is the
easiest way to start with questions like this one.

Your sequence is number A011541, the "Hardy-Ramanujan numbers".

> A. Is R(k) defined for all k?

Apparently so.

> B. What is R(2)?

87539319.

> C. What about R(k) for other values of k?

R(3)=6963472309248.
R(4)=48988659276962496.
R(5)<=8230545258248091551205888.

> D. Is the answer the same if we say "k+1 distinct solutions"
> instead of "more than k distinct solutions.

Probably.

                                        David desJardins
==============================================================================

From: Werner Nickel <nickel@fb04196.mathematik.tu-darmstadt.de>
Subject: Re: 2, 1729, ?
Date: 2 Jan 2001 13:04:10 GMT
Newsgroups: sci.math.research

Edwin Clark <eclark@math.usf.edu> wrote:

> Let k be a positive integer.

> Let R(k) be the smallest positive integer n for which

>                   n = x^3 + y^3

> has more than k distinct soluions x,y in positive integers?

[...]

> B. What is R(2)?

A straight forward exhaustive search in GAP 4 revealed
R(2) = 87,539,319 and that R(3) <> 87,539,319.  The three
solutions are:

	[ 414, 255 ], [ 423, 228 ], [ 436, 167 ]

Other values for n with k=2 are:

119,824,488: [ 428, 346 ], [ 492, 90 ],  [ 493, 11 ]
143,604,279: [ 423, 408 ], [ 460, 359 ], [ 522, 111 ]
175,959,000: [ 525, 315 ], [ 552, 198 ], [ 560, 70 ]

All the best,
Werner Nickel
==============================================================================

From: njas@research.att.com (N. J. A. Sloane)
Subject: Re: 2, 1729, ?
Date: 2 Jan 01 14:01:04 GMT
Newsgroups: sci.math.numberthy

The EIS contains this sequence:

%I A011541
%S A011541 2,1729,87539319,6963472309248,48988659276962496
%N A011541 Taxi-cab or Hardy-Ramanujan numbers: the smallest number that is the sum of 2 cubes in n ways (an infinite sequence).
%D A011541 R. K. Guy, Unsolved Problems in Number Theory, D1.
%D A011541 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 165 and 189.
%H A011541 <a href="http://www.research.att.com/~njas/sequences/JIS/index.html#P98.1.6">D. W. Wilson, The Fifth Taxicab Number is 48988659276962496,</a>J. Integer Sequences, Vol. 2, 1999, #9.
%H A011541 <a href="http://pobox.com/~djb/papers/sortedsums.dvi">5th term computed independently (and slightly later) by D. J. Bernstein</a>
%H A011541 <a href="http://mathworld.wolfram.com/CubicNumber.html">Link to a section of Eric Weisstein's World of Mathematics.</a>
%H A011541 <a href="http://mathworld.wolfram.com/TaxicabNumber.html">Link to a section of Eric Weisstein's World of Mathematics.</a>
%Y A011541 Cf. A023050, A003826, A001235.
%K A011541 nonn,nice,hard
%O A011541 1,1
%A A011541 njas, Robert G. Wilson v (rgwv@kspaint.com)
%E A011541 dww reports a(6) <= 8230545258248091551205888.

NJAS
==============================================================================

From: Henayni@hotmail.com (Matt Herman)
Subject: Re: 2, 1729, ?
Date: 2 Jan 01 14:01:04 GMT
Newsgroups: sci.math.numberthy

Hi,

From

-
Taxicabs and Sums of Two Cubes
J. Silverman
American Mathematical Monthly, Volume 100, Issue 4 (Apr., 1993), 331-340.
-

we have the following results.
R(2) = 87539319 = 436^3 + 167^3 = 423^3 + 228^3 = 414^3 + 255^3
If you want R(2) to be cube free, the smallest representation is
15170835645 = 517^3 + 2468^3 = 709^3 + 2456^3 = 1733^3 + 2152^3.

For the minimal integer with four representations, we have 26059452841000 =
29620^3 + 4170^3 = 28810^3 + 12900^3
= 28423^3 + 14577^3 = 24940^3 + 21930^3.

However, as of 1993, there is no known cube-free integer that has four
different representations (even when negative x,y are allowed).

The main result of the paper is a answer to A.
Given any integer N, there exists a positive integer A for which the
equation X^3 + Y^3 = A has at least N solutions in integers.
From this, it can be easily proven that there are infinitely many positive
integer solutions.

The method of proof is somewhat constructive -- start with a rational
solution to X^3 + Y^3 = 7, P = (2, -1). Then, we have a result that the
sequence P, 2P, .. , which never repeats. So take the first N. Then nP has
the form (a_n/d_n, b_n/d_n). So multiply the original equation by the
product of the d_i (i=1,..n) = B. Then the equation X^3 + Y^3 = 7B^3 has at
least N solutions in integers, namely B*(a_n/d_n, b_n,d_n), 0<n<N+1.We then
prove that infinitely many of the nP have positive coordinate, this proves
the second part of the theorem.

We also have the following result, due to K. Mahler:

There is a constant c > 0 such that for infinitely many positive integeres
A, the number of positive integer solutions to the equation X^3 + Y^3 = A
exceeds c * (log A)^(1/3).

I hope this helps. You could probably get this paper on JSTOR.

Happy New Year!

----- Original Message -----
[deleted --djr]
==============================================================================

From: kohmoto@z2.zzz.or.jp (y.kohmoto)
Subject: a generalized taxi cab
Date: 19 Jan 01 13:02:45 GMT
Newsgroups: sci.math.numberthy

     T(k,m,n) :
         the smallest number such that representable as a sum of m k powers
in n ways.

         ex.
         T(3,2,n) is the same thing as Ta(n) which is known as Taxi cab
Number :
         the smallest  number representable as a sum of two cubes in n ways.
         for n<=5 , they are known.

    [The smallest records for T(3,m,n)]
         T(3,2,6)=Ta(6)<=8230545258248091551205888
= 11239317^3 + 201891435^3
= 17781264^3 + 201857064^3
= 63273192^3 + 199810080^3
= 85970916^3 + 196567548^3
= 125436328^3 + 184269296^3
= 159363450^3 + 161127942^3
            .      ( David W. Wilson )

    I have two examples.
         T(3,3,16)<=810000^3
=  809730^3+80991^3+9^3
=  807840^3+161856^3+144^3
=  802710^3+242271^3+729^3
=  792720^3+321696^3+2304^3
=  776250^3+399375^3+5625^3
=  766200^3+433800^3+6000^3
=  751680^3+474336^3+11664^3
=  656400^3+628800^3+14400^3
=  796500^3+295500^3+69000^3
=  795000^3+306000^3+69000^3
=  720000^3+540000^3+90000^3
=  782550^3+373725^3+27675^3
=  788355^3+346140^3+15165^3
=  783000^3+371250^3+60750^3
=  724140^3+532980^3+68040^3
=  736306^3+509492^3+6366^3

         T(3,4,15)<=1350000^3
=    1349538^3+136140^3+12360^3+462^3
=    1346888^3+256987^3+15397^3+728^3
=    1346737^3+261085^3+7421^3+221^3
=    1339010^3+390610^3+13034^3+466^3
=    1333215^3+449190^3+12330^3+225^3
=    1285830^3+694124-3+12772^3+762^3
=    1349200^3+163440^3+17640^3+1880^3
=    1346760^3+260280^3+34020^3+540^3
=    1342800^3+339570^3+11070^3+540^3
=    1339160^3+388750^3+34930^3+1880^3
=    1337000^3+412880^3+22660^3+1180^3
=    1330500^3+471870^3+25440^3+1170^3
=    1311240^3+590490^3+14310^3+960^3
=    1235150^3+832050^3+11400^3+1000^3
=    1230230^3+842700^3+29680^3+2010^3



In addition:

5949503719112230284735150781047420975571215969633783660978692570826281101309
189803771999166533150006397466555165181072458877^3
+693167423530203568796466556841646172413753795462810020862370805265266981207
049609344143725917800499174450944129560853125400625152^3
-129576938056899065227232353532280140679420086701403228248146670988068053133
80604081166444397018867983934899325^3
=2^1284

    2^1284 is 2-adically small, so this is almost an example of
a^3+b^3-c^3=0