{"id":2516,"date":"2019-09-11T02:24:42","date_gmt":"2019-09-11T06:24:42","guid":{"rendered":"http:\/\/blogs.ams.org\/beyondreviews\/?p=2516"},"modified":"2019-09-11T02:54:27","modified_gmt":"2019-09-11T06:54:27","slug":"42","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2019\/09\/11\/42\/","title":{"rendered":"42"},"content":{"rendered":"

\"DrawingThe number 42 is famous for its occurrence in The<\/em> Hitchhiker’s Guide to the Galaxy<\/em>. In 2032, Adele might come out with a new album with 42<\/em> as its title.\u00a0 But today, the fame of the number 42 has to do with its representation as a sum of three cubes.\u00a0 <\/p>\n

The problem is to try to represent each positive integer less than or equal to 100 as the sum of three cubes:\u00a0 $x^3 + y^3 + z^3=n$, where $n$ is the given integer between 1 and 100 and $x$, $y$, and $z$ are the integers you need to find.\u00a0 Some choices of $n$ are easy.\u00a0 Some are known to be impossible, such as any $n$ that is equal to 4 or 5 mod 9.\u00a0 Until recently, the answer was known for every $n$ except $n=33$ and $n=42$.\u00a0 Then, in early 2019, Andrew Booker<\/a> announced a representation of<\/a>\u00a033 as three cubes<\/a>, using some good ideas and an enormous amount of computing time.\u00a0 Hurray!\u00a0 Booker was rightly celebrated for his result.\u00a0 See, for example, the write-up in Quanta Magazine<\/a>.\u00a0 But the question remained for one more number: 42.<\/p>\n

To try to crack 42, Booker teamed up with Andrew Sutherland<\/a>.\u00a0 Sutherland had a track record of breaking records using massively parallel computing.\u00a0 Given the amount of time needed to solve 33, it seemed like parallel computing would be necessary (or at least a good idea) for 42.\u00a0 They used Charity Engine<\/a>, a global parallel computer that uses millions of idle personal PCs(*)<\/sup>.\u00a0 After some millions of hours of computer time, Booker and Sutherland came up with the answer:<\/p>\n

$ (-80538738812075974)^3 + (80435758145817515)^3 + (12602123297335631)^3 = 42$<\/p>\n

Ta Dah!<\/p>\n

The problem has a rich history, with lots of powerful ideas that produce integers with quite a few digits whose cubes add up to one- or two-digit numbers.\u00a0 Reviews of a handful of papers on the subject are provided below.\u00a0 There is also a nice video<\/a> from Numberphile<\/a> about the problem, which was posted to YouTube in November 2015, before Booker’s work on either 33 or 42, and a newer video<\/a> posted just a few days ago, that discusses 42.<\/p>\n

\n

(*)<\/sup> The use of\u00a0a global network of otherwise idle computers is also used in the hunt for Mersenne primes. The GIMPS project<\/a> has been finding record large primes this way since 1996.<\/p>\n

\n

MR1850598<\/strong>
\nElkies, Noam D.<\/a>(1-HRV)<\/a><\/span>
\nRational points near curves and small nonzero\u00a0$|x^3-y^2|$<\/span>\u00a0via lattice reduction.<\/span>\u00a0(English summary)<\/span>\u00a0Algorithmic number theory (Leiden, 2000),\u00a0<\/em>33\u201363,
\nLecture Notes in Comput. Sci., 1838,<\/a>\u00a0Springer, Berlin,<\/em>\u00a02000.
\n11D75 (11G50 11H55 11J25)<\/a><\/p>\n

<\/div>\n

The author gives an algorithm for effective calculation of all rational points of small height near a given plane curve $C$<\/span>. Cases treated include the inequality\u00a0$|x^3 + y^3 – z^3| < M \\quad (0 < x \\le y < z < N)$<\/span>. Its solutions can be enumerated in heuristic time\u00a0$\\ll M\\log^CN $<\/span>\u00a0provided\u00a0$M \\gg N$<\/span>, using\u00a0$O(\\log N)$<\/span>\u00a0space. The algorithm is adapted to find all integer solutions of\u00a0$0 < |x^3 – y^2| \\ll x^{1\/2}$<\/span>\u00a0with\u00a0$x < X$<\/span>\u00a0in (rigorous) time\u00a0$O(X^{1\/2} \\log^{O(1)}X)$<\/span>. The strength of the algorithm, which involves linear approximation to the curve and lattice reduction, is illustrated by an example of integers\u00a0$x,y$<\/span>\u00a0with\u00a0$x^{1\/2}\/|x^3 – y^2|$<\/span>\u00a0greater than 46. The previous record was 4.87. The author discusses variations and generalizations. For example, fix an algebraic curve\u00a0$C\/\\Bbb{Q}$<\/span>\u00a0and a divisor\u00a0$D$<\/span>\u00a0on\u00a0$C$<\/span>\u00a0of degree\u00a0$d > 0$<\/span>. One can find the points of\u00a0$C$<\/span>\u00a0whose height relative to\u00a0$D$<\/span>\u00a0is at most\u00a0$H$<\/span> in time $A(\\epsilon)H^{(2\/d)+\\epsilon}$<\/span>. Here\u00a0$A(\\epsilon)$<\/span>\u00a0is effectively computable for each\u00a0$\\epsilon > 0$<\/span>.<\/p>\n

{For the collection containing this paper see\u00a0MR1850596<\/a>.}<\/span><\/p>\n

Reviewed by\u00a0R. C. Baker<\/a><\/span><\/p>\n

\n

MR1146835<\/a>
\nHeath-Brown, D. R.<\/a>(4-OXM)<\/a><\/span>
\nThe density of zeros of forms for which weak approximation fails.<\/span>
\nMath. Comp.<\/em><\/a>\u00a059\u00a0<\/a>(1992),\u00a0<\/a>no. 200,<\/a>\u00a0613\u2013623.
\n11G35 (11D25 11P55)<\/a><\/p>\n

Let\u00a0$f_k(x_1,x_2,x_3,x_4)=x^3_1+x^3_2+x^3_3-kx^3_4$<\/span>. The author shows that if\u00a0${\\bf x}$<\/span>\u00a0is a primitive solution of\u00a0$f_2({\\bf x})=0$<\/span>, then one of\u00a0$x_1,x_2,x_3$<\/span>\u00a0is divisible by 6. Thus\u00a0$f_2$<\/span>\u00a0has no rational zero close to both\u00a0$(0,1,1,1)\\in{\\bf Q}^4_2$<\/span>\u00a0and\u00a0$(1,0,1,1)\\in{\\bf Q}^4_3$<\/span>. Since these are zeros of\u00a0$f_2$<\/span>, the “weak approximation” principle fails for\u00a0$f_2$<\/span>. He obtains an analogous result for\u00a0$f_3$<\/span>.<\/p>\n

Let $R_k(N)=\\#\\{{\\bf x}\\in{\\bf Z}^4\\colon\\;\\max_{i\\leq 3}|x_i|\\leq N$<\/span>,\u00a0${\\bf x}$<\/span>\u00a0primitive,\u00a0$f_k({\\bf x})=0\\}$<\/span>. The author computes\u00a0$R_k(1000)$<\/span>\u00a0$(k=2,3)$<\/span>\u00a0and observes some agreement with the hypothetical Hardy-Littlewood formula (1)\u00a0$R_k(N)\\sim {\\frak S}_kN$<\/span>. Subject to a conjecture on uniform convergence of certain products, he establishes that $\\sum_{K<k\\leq 2K}^*R_k(N)\\sim N\\sum^*_{K<k\\leq 2K}{\\frak S}_k$<\/span>, where\u00a0$\\sum^*$<\/span>\u00a0indicates omission of cubes. Perhaps, despite failure of the weak approximation principle, (1) is correct.<\/p>\n

Reviewed by\u00a0R. C. Baker<\/a><\/span><\/p>\n

\n

MR0000026<\/a>
\nDavenport, H.<\/a>
\nOn Waring’s problem for cubes.<\/span>
\nActa Math.<\/em><\/a>\u00a071,\u00a0<\/strong>(1939). 123\u2013143.
\n10.0X<\/a><\/p>\n

There are very few results concerning Waring’s problem which may be called best possible. One of these is the main theorem of this paper: If\u00a0$E_s(N)$<\/span>\u00a0denotes the number of positive integers not greater than\u00a0$N$<\/span>\u00a0which are not representable as a sum of\u00a0$s$<\/span>\u00a0positive integral cubes, then\u00a0$E_4(N)\/N\\to 0$<\/span>\u00a0as\u00a0$N\\to \\infty$<\/span>. A simple argument shows that the theorem is false if\u00a0$s < 4$<\/span>.<\/p>\n

The new weapon used here has been explained in detail in several papers by the author [C. R. Acad. Sci. Paris 207<\/span>, 1366 (1938); Proc. Roy. Soc. London\u00a0170<\/span>, 293\u2013299 (1939); Ann. of Math.\u00a040<\/span>, 533\u2013536 (1939)]. The remainder of the proof is similar to that given by Landau [Vorlesungen \u00fcber Zahlentheorie, Bd. 1, 235\u2013303] for the third Hardy-Littlewood theorem [Satz 346] as adapted by Davenport and Heilbronn [Proc. London Math. Soc. (2)\u00a043<\/span>, 73\u2013104 (1937)] to the problem of representations by two cubes and one square. Interesting parts of the proof are the use of Poisson’s summation formula in Lemma 7 and the handling of the Singular Series.<\/p>\n

Reviewed by\u00a0R. D. James<\/a><\/span><\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

The number 42 is famous for its occurrence in The Hitchhiker’s Guide to the Galaxy. In 2032, Adele might come out with a new album with 42 as its title.\u00a0 But today, the fame of the number 42 has to … Continue reading →<\/span><\/a><\/p>\n

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