Sums of three cubes

Question, Web Fundamentals.svg 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)

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 to equal such a sum is that 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.

Semi-log plot of solutions of for integer , , and , and . Green bands denote values of proven not to have a solution.

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 casesEdit

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


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

  (discovered[3] by K. Mahler in 1936)


  (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 exist other representations, and other parameterized families of representations, for 1.[6] For 2, the other known representations are[6][7]


However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials in this way.[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


and more than the fact that in this case each of the three cubed numbers must be equal modulo 9.[8][9]

Computational resultsEdit

Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations.[10][11][7][12][13][14][15][16][17][18]Elsenhans & Jahnel (2009) used a method of Noam Elkies (2000) involving lattice reduction to search for all solutions to the Diophantine equation


for positive   at most 1000 and for  ,[17], leaving only 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, and 975 as open problems for  . After Timothy Browning covered the problem on Numberphile, Huisman (2016) extended these searches to   solving the case of 74, with solution


Through these searches, it was discovered that all   that are unequal to 4 or 5 modulo 9 have a solution, with at most two exceptions, 33 and 42.[18]

In 2019, Andrew Booker settled the case   by discovering that


In order to achieve this, Booker exploited an alternative search strategy with running time proportional to   rather than to their maximum,[19] an approach originally suggested by Heath-Brown et al.[20] He also found that


and established that there are no solutions for   or any of the other unresolved   with  .

In September 2019, Andrew Booker and Andrew Sutherland settled the   case, using 1.3 million hours of computing on the Charity Engine global grid to discover that






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


This discovery settled a 65-year old question of Louis J. Mordell that has stimulated much of the research on this problem.[23]

The only remaining unsolved cases up to 1,000 are 114, 390, 579, 627, 633, 732, 921 and 975.[21][24]

Popular interestEdit

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.[25] This was followed six months later by the video "74 is Cracked" with Browning, discussing Huisman's 2016 discovery of a solution for 74.[26] 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.[27][28][29]

Booker's solution for 33 was featured in articles appearing in Quanta Magazine[30] and New Scientist[31], 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".[32] The number 42 has additional popular interest due to its appearance in the 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[33][34] of a solution for 42 received international press coverage, including articles in New Scientist,[35] Popular Mechanics,[36] The Register,[37] Die Zeit,[38] Der Tagesspiegel,[39] Helsingin Sanomat,[40] Der Spiegel,[41] New Zealand Herald,[42] Indian Express,[43] Der Standard,[44] Las Provincias,[45] Nettavisen,[46] Digi24,[47] and BBC World Service.[48] Popular Mechanics named the solution for 42 as one of the "10 Biggest Math Breakthroughs of 2019".[49]

The resolution of Mordell's question by Booker and Sutherland a few weeks later sparked another round of news coverage.[22][50][51][52][53][54][55]

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.[56]

Solvability and decidabilityEdit

In 1992, Roger Heath-Brown conjectured that every   unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes.[57] The case   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.[58] 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   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.[57]


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.[59] It is conjectured that the representable numbers have positive natural density.[60][61] This remains unknown, but Trevor Wooley has shown that   of the numbers from   to   have such representations.[62][63][64] The density is at most  .[1]

Every integer can be represented as a sum of three cubes of rational numbers (rather than as a sum of cubes of integers).[65][66]


  1. ^ a b 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  ], Matematicheskii Sbornik (in Russian), 26 (4): 622–624, JFM 39.0259.02
  5. ^ a b c 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. ^ a b Avagyan, Armen; Dallakyan, Gurgen (2018), A new method in the problem of three cubes, arXiv:1802.06776, doi:10.13189/ujcmj.2017.050301 (inactive 2020-09-09)
  7. ^ a b Heath-Brown, D. R.; Lioen, W. M.; te Riele, H. J. J. (1993), "On solving the Diophantine equation   on a vector computer", Mathematics of Computation, 61 (203): 235–244, Bibcode:1993MaCom..61..235H, doi:10.2307/2152950, JSTOR 2152950, MR 1202610
  8. ^ Mordell, L.J. (1953), "On the integer solutions of the equation  ", Journal of the London Mathematical Society, Second Series, 28: 500–510, doi:10.1112/jlms/s1-28.4.500, MR 0056619
  9. ^ 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  ", Mathematics of Computation, 44 (169): 265–266, doi:10.2307/2007811, JSTOR 2007811, MR 0771049, S2CID 121727002.
  10. ^ Miller, J. C. P.; Woollett, M. F. C. (1955), "Solutions of the Diophantine equation  ", Journal of the London Mathematical Society, Second Series, 30: 101–110, doi:10.1112/jlms/s1-30.1.101, MR 0067916
  11. ^ Gardiner, V. L.; Lazarus, R. B.; Stein, P. R. (1964), "Solutions of the diophantine equation  ", Mathematics of Computation, 18 (87): 408–413, doi:10.2307/2003763, JSTOR 2003763, MR 0175843
  12. ^ Conn, W.; Vaserstein, L. N. (1994), "On sums of three integral cubes", The Rademacher legacy to mathematics (University Park, PA, 1992), Contemporary Mathematics, 166, Providence, Rhode Island: American Mathematical Society, pp. 285–294, doi:10.1090/conm/166/01628, MR 1284068
  13. ^ Bremner, Andrew (1995), "On sums of three cubes", Number theory (Halifax, NS, 1994), CMS Conference Proceedings, 15, Providence, Rhode Island: American Mathematical Society, pp. 87–91, MR 1353923
  14. ^ Koyama, Kenji; Tsuruoka, Yukio; Sekigawa, Hiroshi (1997), "On searching for solutions of the Diophantine equation  ", Mathematics of Computation, 66 (218): 841–851, doi:10.1090/S0025-5718-97-00830-2, MR 1401942
  15. ^ Elkies, Noam D. (2000), "Rational points near curves and small nonzero   via lattice reduction", Algorithmic number theory (Leiden, 2000), Lecture Notes in Computer Science, 1838, Springer, Berlin, pp. 33–63, arXiv:math/0005139, doi:10.1007/10722028_2, MR 1850598, S2CID 40620586
  16. ^ 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
  17. ^ a b 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
  18. ^ a b Huisman, Sander G. (2016), Newer sums of three cubes, arXiv:1604.07746
  19. ^ Booker, Andrew R. (2019), "Cracking the problem with 33", Research in Number Theory, 5 (26), doi:10.1007/s40993-019-0162-1, MR 3983550
  20. ^ Heath-Brown, D. R.; Lioen, W.M.; te Riele, H.J.J (1993), "On solving the Diophantine equation   on a vector computer", Mathematics of Computation, 61 (203): 235–244, Bibcode:1993MaCom..61..235H, doi:10.2307/2152950, JSTOR 2152950, MR 1202610
  21. ^ a b c Booker, Andrew R.; Sutherland, Andrew V. (2020), On a question of Mordell, arXiv:2007.01209
  22. ^ a b Lu, Donna (September 18, 2019), "Mathematicians find a completely new way to write the number 3", New Scientist
  23. ^ Mordell, L.J. (1953), "On the Integer Solutions of the Equation  ", Journal of the London Mathematical Society, 28: 500–510, doi:10.1112/jlms/s1-28.4.500, MR 0056619
  24. ^ Houston, Robin (September 6, 2019), "42 is the answer to the question 'what is (-80538738812075974)3 + 804357581458175153 + 126021232973356313?'", The Aperiodical
  25. ^ Haran, Brady (November 6, 2015), The uncracked problem with 33, Numberphile
  26. ^ Haran, Brady (May 31, 2016), 74 is cracked, Numberphile
  27. ^ Haran, Brady (March 12, 2019), 42 is the new 33, Numberphile
  28. ^ Haran, Brady (September 6, 2019), The mystery of 42 is solved, Numberphile
  29. ^ Haran, Brady (September 24, 2019), 3 as the sum of 3 cubes, Numberphile
  30. ^ Pavlus, John (March 10, 2019), "Sum-of-Three-Cubes Problem Solved for 'Stubborn' Number 33", Quanta Magazine
  31. ^ Lu, Donna (March 14, 2019), "Mathematician cracks centuries-old problem about the number 33", New Scientist
  32. ^ Georgiou, Aristos (April 3, 2019), "The uncracked problem with 33: Mathematician solves 64-year-old 'Diophantine puzzle'", Newsweek
  33. ^ Sum of three cubes for 42 finally solved – using real life planetary computer, University of Bristol, September 6, 2019
  34. ^ 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
  35. ^ Lu, Donna (September 6, 2019), "Mathematicians crack elusive puzzle involving the number 42", New Scientist
  36. ^ Grossman, David (September 6, 2019), "After 65 Years, Supercomputers Finally Solve This Unsolvable Math Problem", Popular Mechanics
  37. ^ Quach, Katyanna (September 7, 2019), "Finally! A solution to 42 – the Answer to the Ultimate Question of Life, The Universe, and Everything", The Register
  38. ^ "Matheproblem um die Zahl 42 geknackt", Die Zeit, September 16, 2019
  39. ^ "Das Matheproblem um die Zahl 42 ist geknackt", Der Tagesspiegel, September 16, 2019
  40. ^ Kivimäki, Antti (September 18, 2019), "Matemaatikkojen vaikea laskelma tuotti vihdoin kaivatun luvun 42", Helsingin Sanomat
  41. ^ "Matheproblem um die 42 geknackt", Der Spiegel, September 16, 2019
  42. ^ "Why the number 42 is the answer to life, the universe and everything", New Zealand Herald, September 9, 2019
  43. ^ Firaque, Kabir (September 20, 2019), "Explained: How a 65-year-old maths problem was solved", Indian Express
  44. ^ Taschwer, Klaus (September 15, 2019), "Endlich: Das Rätsel um die Zahl 42 ist gelöst", Der Standard
  45. ^ "Matemáticos resuelven el enigma del número 42 planteado hace 65 años", Las Provincias, September 18, 2019
  46. ^ Wærstad, Lars (October 10, 2019), "Supermaskin har løst over 60 år gammel tallgåte", Nettavisen
  47. ^ "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
  48. ^ Paul, Fernanda (September 12, 2019), "Enigma de la suma de 3 cubos: matemáticos encuentran la solución final después de 65 años", BBC News Mundo
  49. ^ Linkletter, Dave (December 27, 2019), "The 10 Biggest Math Breakthroughs of 2019", Popular Mechanics
  50. ^ Mandelbaum, Ryan F. (September 18, 2019), "Mathematicians No Longer Stumped by the Number 3", Gizmodo
  51. ^ "42:n ongelman ratkaisijat löysivät ratkaisun myös 3:lle", Tiede, September 23, 2019
  52. ^ Kivimäki, Antti (September 22, 2019), "Numeron 42 ratkaisseet matemaatikot yllättivät: Löysivät myös luvulle 3 kauan odotetun ratkaisun", Helsingin Sanomat
  53. ^ Jesus Poblacion, Alfonso (October 3, 2019), "Matemáticos encuentran una nueva forma de llegar al número 3", El Diario Vasco
  54. ^ Honner, Patrick (November 5, 2019), "Why the Sum of Three Cubes Is a Hard Math Problem", Quanta Magazine
  55. ^ D'Souza, Dilip (November 28, 2019), "Waste not, there's a third way to make cubes", LiveMint
  56. ^ Booker, Andrew R. (July 4, 2020), 33 and all that, Algorithmic Number Theory Symposium
  57. ^ a b 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
  58. ^ Poonen, Bjorn (2008), "Undecidability in number theory" (PDF), Notices of the American Mathematical Society, 55 (3): 344–350, MR 2382821
  59. ^ Dickson, Leonard Eugene (1920), History of the Theory of Numbers, Vol. II: Diophantine Analysis, Carnegie Institution of Washington, p. 717
  60. ^ 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
  61. ^ 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, 4076, Berlin: Springer, pp. 141–155, doi:10.1007/11792086_11, MR 2282921
  62. ^ 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
  63. ^ 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
  64. ^ Wooley, Trevor D. (2015), "Sums of three cubes, II", Acta Arithmetica, 170 (1): 73–100, doi:10.4064/aa170-1-6, MR 3373831, S2CID 119155786
  65. ^ 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
  66. ^ 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 linksEdit