Sums of three cubes

Unsolved problem in mathematics:

Is there a number that is not 4 or 5 modulo 9 and that cannot be expressed as a sum of three cubes?

(more unsolved problems in mathematics)

Semi-log plot of solutions of ${\displaystyle x^{3}+y^{3}+z^{3}=n}$ for integer ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$, and ${\displaystyle 0\leq n\leq 100}$. Green bands denote values of ${\displaystyle n}$ proven not to have a solution.

In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer ${\displaystyle n}$ to equal such a sum is that ${\displaystyle n}$ cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. ^[1] It is unknown whether this necessary condition is sufficient.

Variations of the problem include sums of non-negative cubes and sums of rational cubes. All integers have a representation as a sum of rational cubes, but it is unknown whether the sums of non-negative cubes form a set with non-zero natural density.

Small cases

A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's Last Theorem for the exponent three, as one of the three cubes would have the opposite sign as the other two and its negation would equal the sum of the other two. Therefore, by Leonhard Euler's proof of that case of Fermat's last theorem, ^[2] there are only the trivial solutions

${\displaystyle a^{3}+(-a)^{3}+0^{3}=0.}$

For representations of 1 and 2, there are infinite families of solutions

${\displaystyle (9b^{4})^{3}+(3b-9b^{4})^{3}+(1-9b^{3})^{3}=1}$ (discovered ^[3] by K. Mahler in 1936)

and

${\displaystyle (1+6c^{3})^{3}+(1-6c^{3})^{3}+(-6c^{2})^{3}=2}$ (discovered ^[4] by A.S. Verebrusov in 1908, quoted by L.J. Mordell ^[5] ).

These can be scaled to obtain representations for any cube or any number that is twice a cube. ^[5] There are also other known representations of 2 that are not given by these infinite families: ^[6]

${\displaystyle 1\ 214\ 928^{3}+3\ 480\ 205^{3}+(-3\ 528\ 875)^{3}=2,}$

${\displaystyle 37\ 404\ 275\ 617^{3}+(-25\ 282\ 289\ 375)^{3}+(-33\ 071\ 554\ 596)^{3}=2,}$

${\displaystyle 3\ 737\ 830\ 626\ 090^{3}+1\ 490\ 220\ 318\ 001^{3}+(-3\ 815\ 176\ 160\ 999)^{3}=2.}$

However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials as above. ^[5] Even in the case of representations of 3, Louis J. Mordell wrote in 1953 "I do not know anything" more than its small solutions

${\displaystyle 1^{3}+1^{3}+1^{3}=4^{3}+4^{3}+(-5)^{3}=3}$

and the fact that each of the three cubed numbers must be equal modulo 9. ^{[7] [8]}

Computational results

Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations. ^{[9] [10] [6] [11] [12] [13] [14] [15] [16] [17]} Elsenhans & Jahnel (2009) used a method of NoamElkies  ( 2000 ) involving lattice reduction to search for all solutions to the Diophantine equation

${\displaystyle x^{3}+y^{3}+z^{3}=n}$

for positive ${\displaystyle n}$ at most 1000 and for ${\displaystyle \max(|x|,|y|,|z|)<10^{14}}$, ^[16] leaving only 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975 as open problems in 2009 for ${\displaystyle n\leq 1000}$, and 192, 375, and 600 remain with no primitive solutions (i.e. ${\displaystyle \gcd(x,y,z)=1}$). After Timothy Browning covered the problem on Numberphile in 2016, Huisman (2016) extended these searches to ${\displaystyle \max(|x|,|y|,|z|)<10^{15}}$ solving the case of 74, with solution

${\displaystyle 74=(-284\ 650\ 292\ 555\ 885)^{3}+66\ 229\ 832\ 190\ 556^{3}+283\ 450\ 105\ 697\ 727^{3}.}$

Through these searches, it was discovered that all ${\displaystyle n<100}$ that are unequal to 4 or 5 modulo 9 have a solution, with at most two exceptions, 33 and 42. ^[17]

However, in 2019, Andrew Booker settled the case ${\displaystyle n=33}$ by discovering that

${\displaystyle 33=8\ 866\ 128\ 975\ 287\ 528^{3}+(-8\ 778\ 405\ 442\ 862\ 239)^{3}+(-2\ 736\ 111\ 468\ 807\ 040)^{3}.}$

In order to achieve this, Booker exploited an alternative search strategy with running time proportional to ${\displaystyle \min(|x|,|y|,|z|)}$ rather than to their maximum, ^[18] an approach originally suggested by Heath-Brown et al. ^[19] He also found that

${\displaystyle 795=(-14\ 219\ 049\ 725\ 358\ 227)^{3}+14\ 197\ 965\ 759\ 741\ 571^{3}+2\ 337\ 348\ 783\ 323\ 923^{3},}$

and established that there are no solutions for ${\displaystyle n=42}$ or any of the other unresolved ${\displaystyle n\leq 1000}$ with ${\displaystyle |z|\leq 10^{16}}$.

Shortly thereafter, in September 2019, Booker and Andrew Sutherland finally settled the ${\displaystyle n=42}$ case, using 1.3 million hours of computing on the Charity Engine global grid to discover that

${\displaystyle 42=(-80\ 538\ 738\ 812\ 075\ 974)^{3}+80\ 435\ 758\ 145\ 817\ 515^{3}+12\ 602\ 123\ 297\ 335\ 631^{3},}$

as well as solutions for several other previously unknown cases including ${\displaystyle n=165}$ and ${\displaystyle 579}$ for ${\displaystyle n\leq 1000}$. ^[20]

Booker and Sutherland also found a third representation of 3 using a further 4 million computer-hours on Charity Engine:

${\displaystyle 3=569\ 936\ 821\ 221\ 962\ 380\ 720^{3}+(-569\ 936\ 821\ 113\ 563\ 493\ 509)^{3}+(-472\ 715\ 493\ 453\ 327\ 032)^{3}.}$ ^{[20] [21]}

This discovery settled a 65-year-old question of Louis J. Mordell that has stimulated much of the research on this problem. ^[7]

While presenting the third representation of 3 during his appearance in a video on the Youtube channel Numberphile, Booker also presented a representation for 906:

${\displaystyle 906=(-74\ 924\ 259\ 395\ 610\ 397)^{3}+72\ 054\ 089\ 679\ 353\ 378^{3}+35\ 961\ 979\ 615\ 356\ 503^{3}.}$ ^[22]

The only remaining unsolved cases up to 1,000 are the seven numbers 114, 390, 627, 633, 732, 921, and 975, and there are no known primitive solutions (i.e. ${\displaystyle \gcd(x,y,z)=1}$) for 192, 375, and 600. ^{[20] [23]}

Primitive solutions for n from 1 to 78
n	x	y	z		n	x	y	z
1	9	10	−12		39	117367	134476	−159380
2	1214928	3480205	−3528875		42	12602123297335631	80435758145817515	−80538738812075974
3	1	1	1		43	2	2	3
6	−1	−1	2		44	−5	−7	8
7	0	−1	2		45	2	−3	4
8	9	15	−16		46	−2	3	3
9	0	1	2		47	6	7	−8
10	1	1	2		48	−23	−26	31
11	−2	−2	3		51	602	659	−796
12	7	10	−11		52	23961292454	60702901317	−61922712865
15	−1	2	2		53	−1	3	3
16	−511	−1609	1626		54	−7	−11	12
17	1	2	2		55	1	3	3
18	−1	−2	3		56	−11	−21	22
19	0	−2	3		57	1	−2	4
20	1	−2	3		60	−1	−4	5
21	−11	−14	16		61	0	−4	5
24	−2901096694	−15550555555	15584139827		62	2	3	3
25	−1	−1	3		63	0	−1	4
26	0	−1	3		64	−3	−5	6
27	−4	−5	6		65	0	1	4
28	0	1	3		66	1	1	4
29	1	1	3		69	2	−4	5
30	−283059965	−2218888517	2220422932		70	11	20	−21
33	−2736111468807040	−8778405442862239	8866128975287528		71	−1	2	4
34	−1	2	3		72	7	9	−10
35	0	2	3		73	1	2	4
36	1	2	3		74	66229832190556	283450105697727	−284650292555885
37	0	−3	4		75	4381159	435203083	−435203231
38	1	−3	4		78	26	53	−55

Popular interest

The sums of three cubes problem has been popularized in recent years by Brady Haran, creator of the YouTube channel Numberphile, beginning with the 2015 video "The Uncracked Problem with 33" featuring an interview with Timothy Browning. ^[24] This was followed six months later by the video "74 is Cracked" with Browning, discussing Huisman's 2016 discovery of a solution for 74. ^[25] In 2019, Numberphile published three related videos, "42 is the new 33", "The mystery of 42 is solved", and "3 as the sum of 3 cubes", to commemorate the discovery of solutions for 33, 42, and the new solution for 3. ^{[26] [27] [22]}

Booker's solution for 33 was featured in articles appearing in Quanta Magazine ^[28] and New Scientist ^[29] , as well as an article in Newsweek in which Booker's collaboration with Sutherland was announced: "...the mathematician is now working with Andrew Sutherland of MIT in an attempt to find the solution for the final unsolved number below a hundred: 42". ^[30] The number 42 has additional popular interest due to its appearance in the 1979 Douglas Adams science fiction novel The Hitchhiker's Guide to the Galaxy as the answer to The Ultimate Question of Life, the Universe, and Everything.

Booker and Sutherland's announcements ^{[31] [32]} of a solution for 42 received international press coverage, including articles in New Scientist, ^[33] Scientific American , ^[34] Popular Mechanics , ^[35] The Register , ^[36] Die Zeit , ^[37] Der Tagesspiegel , ^[38] Helsingin Sanomat , ^[39] Der Spiegel , ^[40] New Zealand Herald , ^[41] Indian Express , ^[42] Der Standard , ^[43] Las Provincias , ^[44] Nettavisen, ^[45] Digi24, ^[46] and BBC World Service. ^[47] Popular Mechanics named the solution for 42 as one of the "10 Biggest Math Breakthroughs of 2019". ^[48]

The resolution of Mordell's question by Booker and Sutherland a few weeks later sparked another round of news coverage. ^{[21] [49] [50] [51] [52] [53] [54]}

In Booker's invited talk at the fourteenth Algorithmic Number Theory Symposium he discusses some of the popular interest in this problem and the public reaction to the announcement of solutions for 33 and 42. ^[55]

Solvability and decidability

In 1992, Roger Heath-Brown conjectured that every ${\displaystyle n}$ unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes. ^[56] The case ${\displaystyle n=33}$ of this problem was used by Bjorn Poonen as the opening example in a survey on undecidable problems in number theory, of which Hilbert's tenth problem is the most famous example. ^[57] Although this particular case has since been resolved, it is unknown whether representing numbers as sums of cubes is decidable. That is, it is not known whether an algorithm can, for every input, test in finite time whether a given number has such a representation. If Heath-Brown's conjecture is true, the problem is decidable. In this case, an algorithm could correctly solve the problem by computing ${\displaystyle n}$ modulo 9, returning false when this is 4 or 5, and otherwise returning true. Heath-Brown's research also includes more precise conjectures on how far an algorithm would have to search to find an explicit representation rather than merely determining whether one exists. ^[56]

Variations

A variant of this problem related to Waring's problem asks for representations as sums of three cubes of non-negative integers. In the 19th century, Carl Gustav Jacob Jacobi and collaborators compiled tables of solutions to this problem. ^[58] It is conjectured that the representable numbers have positive natural density. ^{[59] [60]} This remains unknown, but Trevor Wooley has shown that ${\displaystyle \Omega (n^{0.917})}$ of the numbers from ${\displaystyle 1}$ to ${\displaystyle n}$ have such representations. ^{[61] [62] [63]} The density is at most ${\displaystyle \Gamma (4/3)^{3}/6\approx 0.119}$. ^[1]

Every integer can be represented as a sum of three cubes of rational numbers (rather than as a sum of cubes of integers). ^{[64] [65]}

See also

• Sum of four cubes problem, whether every integer is a sum of four cubes
• Euler's sum of powers conjecture § k = 3, relating to cubes that can be written as a sum of three positive cubes
• Plato's number, an ancient text possibly discussing the equation 3³ + 4³ + 5³ = 6³
• Taxicab number, the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways

References

1. Davenport, H. (1939), "On Waring's problem for cubes", Acta Mathematica , 71: 123–143, doi: 10.1007/BF02547752 , MR   0000026
2. Machis, Yu. Yu. (2007), "On Euler's hypothetical proof", Mathematical Notes , 82 (3): 352–356, doi:10.1134/S0001434607090088, MR   2364600, S2CID   121798358
3. Mahler, Kurt (1936), "Note on Hypothesis K of Hardy and Littlewood", Journal of the London Mathematical Society , 11 (2): 136–138, doi:10.1112/jlms/s1-11.2.136, MR   1574761
4. Verebrusov, A. S. (1908), "Объ уравненiи x³ + y³ + z³ = 2u³" [On the equation ${\displaystyle x^{3}+y^{3}+z^{3}=2u^{3}}$], Matematicheskii Sbornik (in Russian), 26 (4): 622–624, JFM   39.0259.02
5. Mordell, L.J. (1942), "On sums of three cubes", Journal of the London Mathematical Society , Second Series, 17 (3): 139–144, doi:10.1112/jlms/s1-17.3.139, MR   0007761
6. Heath-Brown, D. R.; Lioen, W. M.; te Riele, H. J. J. (1993), "On solving the Diophantine equation ${\displaystyle x^{3}+y^{3}+z^{3}=k}$ on a vector computer", Mathematics of Computation , 61 (203): 235–244, Bibcode:1993MaCom..61..235H, doi:10.2307/2152950, JSTOR   2152950, MR   1202610
7. Mordell, L.J. (1953), "On the integer solutions of the equation ${\displaystyle x^{2}+y^{2}+z^{2}+2xyz=n}$", Journal of the London Mathematical Society , Second Series, 28: 500–510, doi:10.1112/jlms/s1-28.4.500, MR   0056619
8. The equality mod 9 of numbers whose cubes sum to 3 was credited to J. W. S. Cassels by Mordell (1953), but its proof was not published until Cassels, J. W. S. (1985), "A note on the Diophantine equation ${\displaystyle x^{3}+y^{3}+z^{3}=3}$", Mathematics of Computation , 44 (169): 265–266, doi:10.2307/2007811, JSTOR   2007811, MR   0771049, S2CID   121727002 .
9. Miller, J. C. P.; Woollett, M. F. C. (1955), "Solutions of the Diophantine equation ${\displaystyle x^{3}+y^{3}+z^{3}=k}$", Journal of the London Mathematical Society , Second Series, 30: 101–110, doi:10.1112/jlms/s1-30.1.101, MR   0067916
10. Gardiner, V. L.; Lazarus, R. B.; Stein, P. R. (1964), "Solutions of the diophantine equation ${\displaystyle x^{3}+y^{3}=z^{3}-d}$", Mathematics of Computation , 18 (87): 408–413, doi:10.2307/2003763, JSTOR   2003763, MR   0175843
11. Conn, W.; Vaserstein, L. N. (1994), "On sums of three integral cubes", The Rademacher legacy to mathematics (University Park, PA, 1992), Contemporary Mathematics, vol. 166, Providence, Rhode Island: American Mathematical Society, pp. 285–294, doi:10.1090/conm/166/01628, MR   1284068
12. Bremner, Andrew (1995), "On sums of three cubes", Number theory (Halifax, NS, 1994), CMS Conference Proceedings, vol. 15, Providence, Rhode Island: American Mathematical Society, pp. 87–91, MR   1353923
13. Koyama, Kenji; Tsuruoka, Yukio; Sekigawa, Hiroshi (1997), "On searching for solutions of the Diophantine equation ${\displaystyle x^{3}+y^{3}+z^{3}=n}$", Mathematics of Computation , 66 (218): 841–851, doi: 10.1090/S0025-5718-97-00830-2 , MR   1401942
14. Elkies, Noam D. (2000), "Rational points near curves and small nonzero ${\displaystyle |x^{3}-y^{2}|}$ via lattice reduction", Algorithmic number theory (Leiden, 2000), Lecture Notes in Computer Science, vol. 1838, Springer, Berlin, pp. 33–63, arXiv: math/0005139 , doi:10.1007/10722028_2, ISBN   978-3-540-67695-9, MR   1850598, S2CID   40620586
15. Beck, Michael; Pine, Eric; Tarrant, Wayne; Yarbrough Jensen, Kim (2007), "New integer representations as the sum of three cubes", Mathematics of Computation , 76 (259): 1683–1690, doi: 10.1090/S0025-5718-07-01947-3 , MR   2299795
16. Elsenhans, Andreas-Stephan; Jahnel, Jörg (2009), "New sums of three cubes", Mathematics of Computation , 78 (266): 1227–1230, doi: 10.1090/S0025-5718-08-02168-6 , MR   2476583
17. Huisman, Sander G. (2016), Newer sums of three cubes, arXiv: 1604.07746
18. Booker, Andrew R. (2019), "Cracking the problem with 33", Research in Number Theory, 5 (26), doi: 10.1007/s40993-019-0162-1 , hdl: 1983/b29fce73-2c20-4c07-9daf-afc04bf269b1 , MR   3983550
19. Heath-Brown, D. R.; Lioen, W.M.; te Riele, H.J.J (1993), "On solving the Diophantine equation ${\displaystyle x^{3}+y^{3}+z^{3}=k}$ on a vector computer", Mathematics of Computation , 61 (203): 235–244, Bibcode:1993MaCom..61..235H, doi:10.2307/2152950, JSTOR   2152950, MR   1202610
20. Booker, Andrew R.; Sutherland, Andrew V. (2021), "On a question of Mordell", Proceedings of the National Academy of Sciences, 118 (11), arXiv: 2007.01209 , doi: 10.1073/pnas.2022377118 , PMC   7980389 , PMID   33692126
21. Lu, Donna (September 18, 2019), "Mathematicians find a completely new way to write the number 3", New Scientist
22. Haran, Brady (September 24, 2019), 3 as the sum of 3 cubes, Numberphile
23. Houston, Robin (September 6, 2019), "42 is the answer to the question 'what is (-80538738812075974)³ + 80435758145817515³ + 12602123297335631³?'", The Aperiodical
24. Haran, Brady (November 6, 2015), The uncracked problem with 33, Numberphile
25. Haran, Brady (May 31, 2016), 74 is cracked, Numberphile
26. Haran, Brady (March 12, 2019), 42 is the new 33, Numberphile
27. Haran, Brady (September 6, 2019), The mystery of 42 is solved, Numberphile
28. Pavlus, John (March 10, 2019), "Sum-of-Three-Cubes Problem Solved for 'Stubborn' Number 33", Quanta Magazine
29. Lu, Donna (March 14, 2019), "Mathematician cracks centuries-old problem about the number 33", New Scientist
30. Georgiou, Aristos (April 3, 2019), "The uncracked problem with 33: Mathematician solves 64-year-old 'Diophantine puzzle'", Newsweek
31. Sum of three cubes for 42 finally solved – using real life planetary computer, University of Bristol, September 6, 2019
32. Miller, Sandi (September 10, 2019), "The answer to life, the universe, and everything: Mathematics researcher Drew Sutherland helps solve decades-old sum-of-three-cubes puzzle, with help from "The Hitchhiker's Guide to the Galaxy."", MIT News, Massachusetts Institute of Technology
33. Lu, Donna (September 6, 2019), "Mathematicians crack elusive puzzle involving the number 42", New Scientist
34. Delahaye, Jean-Paul (September 20, 2020), "For Math Fans: A Hitchhiker's Guide to the Number 42", Scientific American
35. Grossman, David (September 6, 2019), "After 65 Years, Supercomputers Finally Solve This Unsolvable Math Problem", Popular Mechanics
36. Quach, Katyanna (September 7, 2019), "Finally! A solution to 42 – the Answer to the Ultimate Question of Life, The Universe, and Everything", The Register
37. "Matheproblem um die Zahl 42 geknackt", Die Zeit, September 16, 2019
38. "Das Matheproblem um die Zahl 42 ist geknackt", Der Tagesspiegel, September 16, 2019
39. Kivimäki, Antti (September 18, 2019), "Matemaatikkojen vaikea laskelma tuotti vihdoin kaivatun luvun 42", Helsingin Sanomat
40. "Matheproblem um die 42 geknackt", Der Spiegel, September 16, 2019
41. "Why the number 42 is the answer to life, the universe and everything", New Zealand Herald, September 9, 2019
42. Firaque, Kabir (September 20, 2019), "Explained: How a 65-year-old maths problem was solved", Indian Express
43. Taschwer, Klaus (September 15, 2019), "Endlich: Das Rätsel um die Zahl 42 ist gelöst", Der Standard
44. "Matemáticos resuelven el enigma del número 42 planteado hace 65 años", Las Provincias, September 18, 2019
45. Wærstad, Lars (October 10, 2019), "Supermaskin har løst over 60 år gammel tallgåte", Nettavisen
46. "A fost rezolvată problema care le-a dat bătăi de cap matematicienilor timp de 6 decenii. A fost nevoie de 1 milion de ore de procesare", Digi24, September 16, 2019
47. Paul, Fernanda (September 12, 2019), "Enigma de la suma de 3 cubos: matemáticos encuentran la solución final después de 65 años", BBC News Mundo
48. Linkletter, Dave (December 27, 2019), "The 10 Biggest Math Breakthroughs of 2019", Popular Mechanics
49. Mandelbaum, Ryan F. (September 18, 2019), "Mathematicians No Longer Stumped by the Number 3", Gizmodo
50. "42:n ongelman ratkaisijat löysivät ratkaisun myös 3:lle", Tiede , September 23, 2019
51. Kivimäki, Antti (September 22, 2019), "Numeron 42 ratkaisseet matemaatikot yllättivät: Löysivät myös luvulle 3 kauan odotetun ratkaisun", Helsingin Sanomat
52. Jesus Poblacion, Alfonso (October 3, 2019), "Matemáticos encuentran una nueva forma de llegar al número 3", El Diario Vasco
53. Honner, Patrick (November 5, 2019), "Why the Sum of Three Cubes Is a Hard Math Problem", Quanta Magazine
54. D'Souza, Dilip (November 28, 2019), "Waste not, there's a third way to make cubes", LiveMint
55. Booker, Andrew R. (July 4, 2020), 33 and all that, Algorithmic Number Theory Symposium
56. Heath-Brown, D. R. (1992), "The density of zeros of forms for which weak approximation fails", Mathematics of Computation , 59 (200): 613–623, doi: 10.1090/s0025-5718-1992-1146835-5 , JSTOR   2153078, MR   1146835
57. Poonen, Bjorn (2008), "Undecidability in number theory" (PDF), Notices of the American Mathematical Society , 55 (3): 344–350, MR   2382821
58. Dickson, Leonard Eugene (1920), History of the Theory of Numbers, Vol. II: Diophantine Analysis, Carnegie Institution of Washington, p. 717
59. Balog, Antal; Brüdern, Jörg (1995), "Sums of three cubes in three linked three-progressions", Journal für die Reine und Angewandte Mathematik , 1995 (466): 45–85, doi:10.1515/crll.1995.466.45, MR   1353314, S2CID   118818354
60. Deshouillers, Jean-Marc; Hennecart, François; Landreau, Bernard (2006), "On the density of sums of three cubes", in Hess, Florian; Pauli, Sebastian; Pohst, Michael (eds.), Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006, Proceedings, Lecture Notes in Computer Science, vol. 4076, Berlin: Springer, pp. 141–155, doi:10.1007/11792086_11, ISBN   978-3-540-36075-9, MR   2282921
61. Wooley, Trevor D. (1995), "Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour" (PDF), Inventiones Mathematicae , 122 (3): 421–451, doi:10.1007/BF01231451, hdl: 2027.42/46588 , MR   1359599
62. Wooley, Trevor D. (2000), "Sums of three cubes", Mathematika , 47 (1–2): 53–61 (2002), doi:10.1112/S0025579300015710, hdl: 2027.42/152941 , MR   1924487
63. Wooley, Trevor D. (2015), "Sums of three cubes, II", Acta Arithmetica , 170 (1): 73–100, arXiv: 1502.01944 , doi:10.4064/aa170-1-6, MR   3373831, S2CID   119155786
64. Richmond, H. W. (1923), "On analogues of Waring's problem for rational numbers", Proceedings of the London Mathematical Society , Second Series, 21: 401–409, doi:10.1112/plms/s2-21.1.401, MR   1575369
65. Davenport, H.; Landau, E. (1969), "On the representation of positive integers as sums of three cubes of positive rational numbers", Number Theory and Analysis (Papers in Honor of Edmund Landau), New York: Plenum, pp. 49–53, MR   0262198

External links

• Solutions of n = x³ + y³ + z³ for 0 ≤ n ≤ 99, Hisanori Mishima
• threecubes, Daniel J. Bernstein
• Sums of three cubes, Mathpages