D. J. Bernstein
More number-theoretic computations

# threecubes

The threecubes package has not yet been published.

I'm building a list of integer vectors (a,b,c) such that

• a^3+b^3+c^3 is between -10000 and 10000 inclusive;
• a^3+b^3+c^3 is not a cube; and
• 4(a^3+b^3+c^3) is not (a+b)^3 or (a+c)^3 or (b+c)^3.

Some results I've found with Elkies's method, to be merged into the list: 19990726 19990801 20010729. The last computation took about 1.8 10^14 Athlon-900 cycles.

Beck, Pine, Tarrant, and Yarbrough, using a slower method, found a few of these results in 1999. I'm not aware of other recent computations. Older computations are summarized below.

This work was supported by the National Science Foundation under grants DMS-9600083 and DMS-9970409.

Literature:
• 1955 Miller Woollett: solutions for most n below 100.
• 1964 Gardiner Lazarus Stein: solutions for most n below 999, including 87; reported the following unknowns below 1000: 30? 33? 39? 42? 52? 74? 75? 84? 110? 114? 143? 156? 165? 180? 195? 231? 290? 312? 318? 321? 366? 367? 390? 420? 435? 439? 444? 452? 462? 478? 501? 516? 530? 534? 542? 556? 564? 579? 588? 600? 606? 609? 618? 627? 633? 660? 663? 732? 735? 754? 758? 767? 777? 786? 789? 795? 830? 834? 861? 870? 894? 903? 906? 912? 921? 933? 948? 964? 969? 975?
• 1992 Heath-Brown: algorithm for fixed n taking time essentially L for all (a,b,c) with max{|a|,|b|,|c|} below L.
• 1993 Heath-Brown Lioen te Riele: more details of Heath-Brown's method; computation taking about 30 hours on a Cyber 205; solution for 39 (found in 1991).
• 1994 Conn Vaserstein: solutions for 39 (found in 1991), 84, 556, 870, 960.
• 1994 Koyama: solutions for 39 (found in 1991), 143, 180, 231, 312, 321, 367, 439, 462, 516, 542, 556, 660, 663, 754, 777, 870.
• 1995 Bremner: solutions for 75, 600.
• 1995 Lukes: solutions for 110, 435.
• 1996 Elkies: initial presentation of algorithm taking time essentially L for all (a,b,c,n) below L; rediscovered solutions from 1994 Koyama with much smaller computation.
• 1997 Koyama Tsuruoaka Sekigawa: algorithm for fixed n taking time essentially L^2 for all (a,b,c) with min{|a|,|b|,|c|} below L; computation taking about 8 10^15 cycles on an Alpha Server 2100; solutions for 75, 435, 444, 501, 600, 618, 912, 969; reported a solution for 478 discovered 1995 by Jagy; reported the following unknowns below 1000: 30? 33? 42? 52? 74? 114? 156? 165? 195? 290? 318? 366? 390? 420? 452? 530? 534? 564? 579? 588? 606? 609? 627? 633? 732? 735? 758? 767? 786? 789? 795? 830? 834? 861? 894? 903? 906? 921? 933? 948? 964? 975?
• 2000 Elkies: algorithm taking time essentially L for all (a,b,c,n) below L.