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. But today, the fame of the number 42 has to do with its representation as a sum of three cubes.

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

To try to crack 42, Booker teamed up with Andrew Sutherland. Sutherland had a track record of breaking records using massively parallel computing. 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. They used Charity Engine, a global parallel computer that uses millions of idle personal PCs^{(*)}. After some millions of hours of computer time, Booker and Sutherland came up with the answer:

$ (-80538738812075974)^3 + (80435758145817515)^3 + (12602123297335631)^3 = 42$

Ta Dah!

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. Reviews of a handful of papers on the subject are provided below. There is also a nice video from Numberphile about the problem, which was posted to YouTube in November 2015, before Booker’s work on either 33 or 42, and a newer video posted just a few days ago, that discusses 42.

^{(*)} The use of a global network of otherwise idle computers is also used in the hunt for Mersenne primes. The GIMPS project has been finding record large primes this way since 1996.

**MR1850598**

Elkies, Noam D.(1-HRV)

Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction. (English summary) *Algorithmic number theory (Leiden, 2000), *33–63,

Lecture Notes in Comput. Sci., 1838, *Springer, Berlin,* 2000.

11D75 (11G50 11H55 11J25)

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

{For the collection containing this paper see MR1850596.}

Reviewed by R. C. Baker

MR1146835

Heath-Brown, D. R.(4-OXM)

The density of zeros of forms for which weak approximation fails.

*Math. Comp.* 59 (1992), no. 200, 613–623.

11G35 (11D25 11P55)

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

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

Reviewed by R. C. Baker

MR0000026

Davenport, H.

On Waring’s problem for cubes.

*Acta Math.* **71, **(1939). 123–143.

10.0X

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 $E_s(N)$ denotes the number of positive integers not greater than $N$ which are not representable as a sum of $s$ positive integral cubes, then $E_4(N)/N\to 0$ as $N\to \infty$. A simple argument shows that the theorem is false if $s < 4$.

The new weapon used here has been explained in detail in several papers by the author [C. R. Acad. Sci. Paris 207, 1366 (1938); Proc. Roy. Soc. London 170, 293–299 (1939); Ann. of Math. 40, 533–536 (1939)]. The remainder of the proof is similar to that given by Landau [Vorlesungen über Zahlentheorie, Bd. 1, 235–303] for the third Hardy-Littlewood theorem [Satz 346] as adapted by Davenport and Heilbronn [Proc. London Math. Soc. (2) 43, 73–104 (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.

Reviewed by R. D. James