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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)
Semilog plot of solutions of x³ + y³ + z³ = n for integer x, y and z, and n in [0, 100]. Green bands denote where it has been proven that no solution exists.

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.

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)

and

  (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. 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 developed an alternative search strategy with running time proportional to   rather than to their maximum.[19][20] He and Andrew Sutherland then settled the   case in September 2019 using 1.3 million hours of computing on the Charity Engine global grid to discover that

 [21]

and

 [22]

and

 [23]

The same team found a third representation of 3 using a further 4 million compute-hours on Charity Engine:

 [24][25]

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

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.[26] 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.[27] 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.[26]

VariationsEdit

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.[28] It is conjectured that the representable numbers have positive natural density.[29][30] This remains unknown, but Trevor Wooley has shown that   of the numbers from   to   have such representations.[31][32][33] 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).[34][35]

ReferencesEdit

  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
  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 2019-08-16)
  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.
  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
  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. ^ Kalai, Gil (March 9, 2019), Combinatorics and more
  20. ^ Booker, Andrew R. (2019), Cracking the problem with 33 (PDF), University of Bristol, arXiv:1903.04284
  21. ^ a b Houston, Robin (September 6, 2019), "42 is the answer to the question 'what is (-80538738812075974)3 + 804357581458175153 + 126021232973356313?'", The Aperiodical
  22. ^ "Andrew V. Sutherland personal webpage". Retrieved 20 September 2019.
  23. ^ "Andrew V. Sutherland personal webpage". Retrieved 30 September 2019.
  24. ^ Lu, Donna (18 September 2019). "Mathematicians find a completely new way to write the number 3". New Scientist. Retrieved 11 October 2019.
  25. ^ @markmcan (17 September 2019). "Insanely huge sum-of-three cubes for 3 discovered – after 66 year search" (Tweet) – via Twitter.
  26. ^ 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:10.2307/2153078, JSTOR 2153078, MR 1146835
  27. ^ Poonen, Bjorn (2008), "Undecidability in number theory" (PDF), Notices of the American Mathematical Society, 55 (3): 344–350, MR 2382821
  28. ^ Dickson, Leonard Eugene (1920), History of the Theory of Numbers, Vol. II: Diophantine Analysis, Carnegie Institution of Washington, p. 717
  29. ^ 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
  30. ^ 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
  31. ^ 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
  32. ^ Wooley, Trevor D. (2000), "Sums of three cubes", Mathematika, 47 (1–2): 53–61 (2002), doi:10.1112/S0025579300015710, MR 1924487
  33. ^ Wooley, Trevor D. (2015), "Sums of three cubes, II", Acta Arithmetica, 170 (1): 73–100, doi:10.4064/aa170-1-6, MR 3373831
  34. ^ 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
  35. ^ 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

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