Sums of three cubes

From HandWiki
Short description: Problem in number theory
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)
Semi-log plot of solutions of [math]\displaystyle{ x^3+y^3+z^3=n }[/math] for integer [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math], and [math]\displaystyle{ 0\le n\le 100 }[/math]. Green bands denote values of [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ n }[/math] to equal such a sum is that [math]\displaystyle{ n }[/math] 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

[math]\displaystyle{ a^3 + (-a)^3 + 0^3 = 0. }[/math]

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

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

and

[math]\displaystyle{ (1+6c^3)^3+(1-6c^3)^3+(-6c^2)^3=2 }[/math] (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]

[math]\displaystyle{ 1\ 214\ 928^3 + 3\ 480\ 205^3 + (-3\ 528\ 875)^3 = 2, }[/math]
[math]\displaystyle{ 37\ 404\ 275\ 617^3 + (-25\ 282\ 289\ 375)^3 + (-33\ 071\ 554\ 596)^3 = 2, }[/math]
[math]\displaystyle{ 3\ 737\ 830\ 626\ 090^3 + 1\ 490\ 220\ 318\ 001^3 + (-3\ 815\ 176\ 160\ 999)^3 = 2. }[/math]

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

[math]\displaystyle{ 1^3+1^3+1^3=4^3+4^3+(-5)^3=3 }[/math]

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) used a method of Noam Elkies (2000) involving lattice reduction to search for all solutions to the Diophantine equation

[math]\displaystyle{ x^3+y^3+z^3=n }[/math]

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

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

Through these searches, it was discovered that all [math]\displaystyle{ n \lt 100 }[/math] 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 [math]\displaystyle{ n=33 }[/math] by discovering that

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

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

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

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

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

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

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

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

[math]\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. }[/math][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:

[math]\displaystyle{ 906 = (-74\ 924\ 259\ 395\ 610\ 397)^3 + 72\ 054\ 821\ 089\ 679\ 353\ 378^3 + 35\ 961\ 979\ 615\ 356\ 503^3. }[/math][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. [math]\displaystyle{ \gcd(x,y,z)=1 }[/math]) for 192, 375, and 600.[20][23]

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 [math]\displaystyle{ n }[/math] unequal to 4 or 5 modulo 9 has infinitely many representations as sums of three cubes.[56] The case [math]\displaystyle{ n=33 }[/math] 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 [math]\displaystyle{ n }[/math] 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 [math]\displaystyle{ \Omega(n^{0.917}) }[/math] of the numbers from [math]\displaystyle{ 1 }[/math] to [math]\displaystyle{ n }[/math] have such representations.[61][62][63] The density is at most [math]\displaystyle{ \Gamma(4/3)^3/6\approx 0.119 }[/math].[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

References

  1. ↑ 1.0 1.1 "On Waring's problem for cubes", Acta Mathematica 71: 123–143, 1939, doi:10.1007/BF02547752 
  2. ↑ Machis, Yu. Yu. (2007), "On Euler's hypothetical proof", Mathematical Notes 82 (3): 352–356, doi:10.1134/S0001434607090088 
  3. ↑ "Note on Hypothesis K of Hardy and Littlewood", Journal of the London Mathematical Society 11 (2): 136–138, 1936, doi:10.1112/jlms/s1-11.2.136 
  4. ↑ Verebrusov, A. S. (1908), "Объ уравненiи x3 + y3 + z3 = 2u3" (in Russian), Matematicheskii Sbornik 26 (4): 622–624, http://mi.mathnet.ru/msb6615 
  5. ↑ 5.0 5.1 5.2 "On sums of three cubes", Journal of the London Mathematical Society, Second Series 17 (3): 139–144, 1942, doi:10.1112/jlms/s1-17.3.139 
  6. ↑ 6.0 6.1 "On solving the Diophantine equation [math]\displaystyle{ x^3+y^3+z^3=k }[/math] on a vector computer", Mathematics of Computation 61 (203): 235–244, 1993, doi:10.2307/2152950, Bibcode: 1993MaCom..61..235H, https://ir.cwi.nl/pub/5502 
  7. ↑ 7.0 7.1 "On the integer solutions of the equation [math]\displaystyle{ x^2+y^2+z^2+2xyz=n }[/math]", Journal of the London Mathematical Society, Second Series 28: 500–510, 1953, doi:10.1112/jlms/s1-28.4.500 
  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 "A note on the Diophantine equation [math]\displaystyle{ x^3+y^3+z^3=3 }[/math]", Mathematics of Computation 44 (169): 265–266, 1985, doi:10.2307/2007811 .
  9. ↑ "Solutions of the Diophantine equation [math]\displaystyle{ x^3+y^3+z^3=k }[/math]", Journal of the London Mathematical Society, Second Series 30: 101–110, 1955, doi:10.1112/jlms/s1-30.1.101 
  10. ↑ Gardiner, V. L.; Lazarus, R. B.; Stein, P. R. (1964), "Solutions of the diophantine equation [math]\displaystyle{ x^{3}+y^{3}=z^{3}-d }[/math]", Mathematics of Computation 18 (87): 408–413, doi:10.2307/2003763 
  11. ↑ Conn, W. (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 
  12. ↑ 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 
  13. ↑ Koyama, Kenji; Tsuruoka, Yukio; Sekigawa, Hiroshi (1997), "On searching for solutions of the Diophantine equation [math]\displaystyle{ x^3+y^3+z^3=n }[/math]", Mathematics of Computation 66 (218): 841–851, doi:10.1090/S0025-5718-97-00830-2 
  14. ↑ "Rational points near curves and small nonzero [math]\displaystyle{ |x^3-y^2| }[/math] via lattice reduction", Algorithmic number theory (Leiden, 2000), Lecture Notes in Computer Science, 1838, Springer, Berlin, 2000, pp. 33–63, doi:10.1007/10722028_2, ISBN 978-3-540-67695-9 
  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 
  16. ↑ 16.0 16.1 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 
  17. ↑ 17.0 17.1 Huisman, Sander G. (2016), Newer sums of three cubes 
  18. ↑ Booker, Andrew R. (2019), "Cracking the problem with 33", Research in Number Theory 5 (26), doi:10.1007/s40993-019-0162-1 
  19. ↑ "On solving the Diophantine equation [math]\displaystyle{ x^3+y^3+z^3=k }[/math] on a vector computer", Mathematics of Computation 61 (203): 235–244, 1993, doi:10.2307/2152950, Bibcode: 1993MaCom..61..235H, https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1202610-5/ 
  20. ↑ 20.0 20.1 20.2 Booker, Andrew R.; Sutherland, Andrew V. (2021), "On a question of Mordell", Proceedings of the National Academy of Sciences 118 (11), doi:10.1073/pnas.2022377118, PMID 33692126 
  21. ↑ 21.0 21.1 Lu, Donna (September 18, 2019), "Mathematicians find a completely new way to write the number 3", New Scientist, https://www.newscientist.com/article/2216941-mathematicians-find-a-completely-new-way-to-write-the-number-3/ 
  22. ↑ 22.0 22.1 3 as the sum of 3 cubes, Numberphile, September 24, 2019, https://www.youtube.com/watch?v=GXhzZAem7k0 
  23. ↑ Houston, Robin (September 6, 2019), "42 is the answer to the question 'what is (-80538738812075974)3 + 804357581458175153 + 126021232973356313?'", The Aperiodical, https://aperiodical.com/2019/09/42-is-the-answer-to-the-question-what-is-80538738812075974³-80435758145817515³-12602123297335631³/ 
  24. ↑ The uncracked problem with 33, Numberphile, November 6, 2015, https://www.youtube.com/watch?v=wymmCdLdPvM 
  25. ↑ 74 is cracked, Numberphile, May 31, 2016, https://www.youtube.com/watch?v=_-M_3oV75Lw 
  26. ↑ 42 is the new 33, Numberphile, March 12, 2019, https://www.youtube.com/watch?v=ASoz_NuIvP0 
  27. ↑ The mystery of 42 is solved, Numberphile, September 6, 2019, https://www.youtube.com/watch?v=zyG8Vlw5aAw 
  28. ↑ Pavlus, John (March 10, 2019), "Sum-of-Three-Cubes Problem Solved for 'Stubborn' Number 33", Quanta Magazine, https://www.quantamagazine.org/sum-of-three-cubes-problem-solved-for-stubborn-number-33-20190326/ 
  29. ↑ Lu, Donna (March 14, 2019), "Mathematician cracks centuries-old problem about the number 33", New Scientist, https://www.newscientist.com/article/2196549-mathematician-cracks-centuries-old-problem-about-the-number-33/ 
  30. ↑ Georgiou, Aristos (April 3, 2019), "The uncracked problem with 33: Mathematician solves 64-year-old 'Diophantine puzzle'", Newsweek, https://www.newsweek.com/uncracked-problem-mathematician-diophantine-puzzle-1384422 
  31. ↑ Sum of three cubes for 42 finally solved – using real life planetary computer, University of Bristol, September 6, 2019, https://www.bristol.ac.uk/news/2019/september/sum-of-three-cubes-.html 
  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), https://news.mit.edu/2019/answer-life-universe-and-everything-sum-three-cubes-mathematics-0910 
  33. ↑ Lu, Donna (September 6, 2019), "Mathematicians crack elusive puzzle involving the number 42", New Scientist, https://www.newscientist.com/article/2215680-mathematicians-crack-elusive-puzzle-involving-the-number-42/ 
  34. ↑ Delahaye, Jean-Paul (September 20, 2020), "For Math Fans: A Hitchhiker's Guide to the Number 42", Scientific American, https://www.scientificamerican.com/article/for-math-fans-a-hitchhikers-guide-to-the-number-42/ 
  35. ↑ Grossman, David (September 6, 2019), "After 65 Years, Supercomputers Finally Solve This Unsolvable Math Problem", Popular Mechanics, https://www.popularmechanics.com/science/math/a28943849/unsolvable-math-problem/ 
  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, https://www.theregister.co.uk/2019/09/07/three_cubes_problem/ 
  37. ↑ "Matheproblem um die Zahl 42 geknackt", Die Zeit, September 16, 2019, https://www.zeit.de/news/2019-09/16/matheproblem-um-die-zahl-42-geknackt 
  38. ↑ "Das Matheproblem um die Zahl 42 ist geknackt", Der Tagesspiegel, September 16, 2019, https://www.tagesspiegel.de/wissen/die-antwort-auf-alles-das-matheproblem-um-die-zahl-42-ist-geknackt/25021142.html 
  39. ↑ Kivimäki, Antti (September 18, 2019), "Matemaatikkojen vaikea laskelma tuotti vihdoin kaivatun luvun 42", Helsingin Sanomat, https://www.hs.fi/tiede/art-2000006242092.html 
  40. ↑ "Matheproblem um die 42 geknackt", Der Spiegel, September 16, 2019, https://www.spiegel.de/wissenschaft/mensch/zahlentheorie-matheproblem-um-die-zahl-42-geknackt-a-1286989.html 
  41. ↑ "Why the number 42 is the answer to life, the universe and everything", New Zealand Herald, September 9, 2019, https://www.hs.fi/tiede/art-2000006247445.html 
  42. ↑ Firaque, Kabir (September 20, 2019), "Explained: How a 65-year-old maths problem was solved", Indian Express, https://indianexpress.com/article/explained/how-a-65-year-old-maths-problem-was-solved-6011373/ 
  43. ↑ Taschwer, Klaus (September 15, 2019), "Endlich: Das Rätsel um die Zahl 42 ist gelöst", Der Standard, https://www.derstandard.de/story/2000108538325/endlich-das-raetsel-um-die-zahl-42-ist-geloest 
  44. ↑ "Matemáticos resuelven el enigma del número 42 planteado hace 65 años", Las Provincias, September 18, 2019, https://www.lasprovincias.es/sociedad/ciencia/matematicos-resuelven-enigma-20190916222124-nt.html 
  45. ↑ Wærstad, Lars (October 10, 2019), "Supermaskin har løst over 60 år gammel tallgåte", Nettavisen, https://www.nettavisen.no/livsstil/supermaskin-har-lost-over-60-ar-gammel-tallgate/3423843017.html 
  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, https://www.digi24.ro/stiri/sci-tech/descoperiri/a-fost-rezolvata-problema-care-le-a-dat-batai-de-cap-matematicienilor-timp-de-6-decenii-a-fost-nevoie-de-1-milion-de-ore-de-procesare-1187754 
  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, https://www.bbc.com/mundo/noticias-49675503 
  48. ↑ Linkletter, Dave (December 27, 2019), "The 10 Biggest Math Breakthroughs of 2019", Popular Mechanics, https://www.popularmechanics.com/science/math/g30346822/biggest-math-breakthroughs-2019/ 
  49. ↑ Mandelbaum, Ryan F. (September 18, 2019), "Mathematicians No Longer Stumped by the Number 3", Gizmodo, https://gizmodo.com/mathematicians-no-longer-stumped-by-the-number-3-1838221738 
  50. ↑ "42:n ongelman ratkaisijat löysivät ratkaisun myös 3:lle", Tiede, September 23, 2019, https://www.tiede.fi/artikkeli/uutiset/42n-ongelman-ratkaisijat-loysivat-ratkaisun-myos-3lle 
  51. ↑ Kivimäki, Antti (September 22, 2019), "Numeron 42 ratkaisseet matemaatikot yllättivät: Löysivät myös luvulle 3 kauan odotetun ratkaisun", Helsingin Sanomat, https://www.hs.fi/tiede/art-2000006247445.html 
  52. ↑ Jesus Poblacion, Alfonso (October 3, 2019), "Matemáticos encuentran una nueva forma de llegar al número 3", El Diario Vasco, https://www.diariovasco.com/sociedad/ciencia/matematicos-encuentran-nueva-20191003130121-nt.html 
  53. ↑ Honner, Patrick (November 5, 2019), "Why the Sum of Three Cubes Is a Hard Math Problem", Quanta Magazine, https://www.quantamagazine.org/why-the-sum-of-three-cubes-is-a-hard-math-problem-20191105/ 
  54. ↑ D'Souza, Dilip (November 28, 2019), "Waste not, there's a third way to make cubes", LiveMint, https://www.livemint.com/science/news/waste-not-there-s-a-third-way-to-make-cubes-11574958211678.html 
  55. ↑ Booker, Andrew R. (July 4, 2020), 33 and all that, Algorithmic Number Theory Symposium, https://www.youtube.com/watch?v=so6VBxWwWVI 
  56. ↑ 56.0 56.1 "The density of zeros of forms for which weak approximation fails", Mathematics of Computation 59 (200): 613–623, 1992, doi:10.1090/s0025-5718-1992-1146835-5 
  57. ↑ "Undecidability in number theory", Notices of the American Mathematical Society 55 (3): 344–350, 2008, https://www.ams.org/notices/200803/tx080300344p.pdf 
  58. ↑ History of the Theory of Numbers, Vol. II: Diophantine Analysis, Carnegie Institution of Washington, 1920, p. 717, https://archive.org/details/historyoftheoryo02dickuoft/page/716 
  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 
  60. ↑ Hess, Florian; Pauli, Sebastian; Pohst, Michael, eds. (2006), "On the density of sums of three cubes", 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, ISBN 978-3-540-36075-9 
  61. ↑ "Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour", Inventiones Mathematicae 122 (3): 421–451, 1995, doi:10.1007/BF01231451, https://deepblue.lib.umich.edu/bitstream/2027.42/46588/1/222_2005_Article_BF01231451.pdf 
  62. ↑ "Sums of three cubes", Mathematika 47 (1–2): 53–61 (2002), 2000, doi:10.1112/S0025579300015710 
  63. ↑ "Sums of three cubes, II", Acta Arithmetica 170 (1): 73–100, 2015, doi:10.4064/aa170-1-6 
  64. ↑ "On analogues of Waring's problem for rational numbers", Proceedings of the London Mathematical Society, Second Series 21: 401–409, 1923, doi:10.1112/plms/s2-21.1.401 
  65. ↑ "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, 1969, pp. 49–53 

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