Sums of three cubes

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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 no solution is proven to exist. The purple line denotes 42, the remaining n < 100 for which a solution has not been found.

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. An obvious 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 cases[edit]

A nontrivial representation of 0 as a sum of three cubes would give a counterexample to Fermat's last theorem for the exponent three. For, 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

and

These can be scaled to obtain representations for any cube or any number that is twice a cube.[3][4] There exist other representations, and other parameterized families of representations, for 1.[5] For 2, the other known representations are[5][6]

However, 1 and 2 are the only numbers with representations that can be parameterized by quartic polynomials in this way.[7] 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] As of March 2019 these remain the only known representations of 3.[10]

Computational results[edit]

Since 1955, and starting with the instigation of Mordell, many authors have implemented computational searches for these representations.[11][12][6][13][14][15][16][17][18][19] 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 ,[18] and Huisman (2016) extended these searches to . Through these searches, it was discovered that all that are unequal to 4 or 5 modulo 9 have a solution, with two exceptions, 33 and 42.[19]

Most recently, 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. After this discovery, the only remaining two-digit value of for which the problem remains unknown is .[20][10]

Solvability and decidability[edit]

In 1992, Roger Heath-Brown conjectured that every unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes.[21] 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.[22] However, 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 easily 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.[21]

Variations[edit]

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.[23] It is conjectured that the representable numbers have positive natural density.[24][25] This, also, remains unknown, but Trevor Wooley has shown that of the numbers from to have such representations.[26][27][28] 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).[29][30]

References[edit]

  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. ^ 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
  4. ^ 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
  5. ^ a b Avagyan, Armen; Dallakyan, Gurgen (2018), A new method in the problem of three cubes, arXiv:1802.06776
  6. ^ 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, MR 1202610
  7. ^ Mordell, L. J. (1942), "On sums of three cubes", Journal of the London Mathematical Society, Second Series, 17: 139–144, doi:10.1112/jlms/s1-17.3.139, MR 0007761
  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, MR 0771049.
  10. ^ a b Booker, Andrew R. (2019), Cracking the problem with 33 (PDF), University of Bristol, arXiv:1903.04284
  11. ^ 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
  12. ^ Gardiner, V. L.; Lazarus, R. B.; Stein, P. R. (1964), "Solutions of the diophantine equation ", Mathematics of Computation, 18: 408–413, doi:10.2307/2003763, MR 0175843
  13. ^ 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
  14. ^ 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
  15. ^ 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
  16. ^ 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, doi:10.1007/10722028_2, MR 1850598
  17. ^ 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
  18. ^ 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
  19. ^ a b Huisman, Sander G. (2016), Newer sums of three cubes, arXiv:1604.07746
  20. ^ Kalai, Gil (March 9, 2019), Combinatorics and more
  21. ^ 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, MR 1146835
  22. ^ Poonen, Bjorn (2008), "Undecidability in number theory" (PDF), Notices of the American Mathematical Society, 55 (3): 344–350, MR 2382821
  23. ^ Dickson, Leonard Eugene (1920), History of the Theory of Numbers, Vol. II: Diophantine Analysis, Carnegie Institution of Washington, p. 717
  24. ^ Balog, Antal; Brüdern, Jörg (1995), "Sums of three cubes in three linked three-progressions", Journal für die Reine und Angewandte Mathematik, 466: 45–85, doi:10.1515/crll.1995.466.45, MR 1353314
  25. ^ 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
  26. ^ Wooley, Trevor D. (1995), "Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour", Inventiones Mathematicae, 122 (3): 421–451, doi:10.1007/BF01231451, MR 1359599
  27. ^ Wooley, Trevor D. (2000), "Sums of three cubes", Mathematika, 47 (1–2): 53–61 (2002), doi:10.1112/S0025579300015710, MR 1924487
  28. ^ Wooley, Trevor D. (2015), "Sums of three cubes, II", Acta Arithmetica, 170 (1): 73–100, doi:10.4064/aa170-1-6, MR 3373831
  29. ^ 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
  30. ^ 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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