Abstract
Inspired by the Numberphile video “The uncracked problem with 33” by Browning and Brady Haran (https://youtu.be/wymmCdLdPvM), we investigate solutions to \(x^3+y^3+z^3=k\) for a few small values of k. We find the first known solutions for \(k=33\) and \(k=795\).
Keywords
Sums of three cubes Diophantine equations Hasse principle1 Introduction
Our strategy is similar to some earlier approaches (see especially [3, 10, 12] and [1]), and is based on the observation that in any solution, \(kz^3=x^3+y^3\) has \(x+y\) as a factor. Our main contribution over the earlier investigations is to note that with some timespace tradeoffs, the running time is very nearly linear in the height bound, and it is quite practical when implemented on modern 64bit computers.
2 Methodology
For ease of presentation, we will assume that \(k\equiv \pm 3\pmod {9}\); note that this holds for all k in (2). Since the basic algorithm described above is reasonable for finding small solutions, we will assume henceforth that \(z>\sqrt{k}\). Also, if we specialize (1) to solutions with \(y=z\), then we get the Thue equation \(x^3+2y^3=k\), which is efficiently solvable. Using the Thue solver in PARI/GP [18], we verify that no such solutions exist for the k in (2). Hence we may further assume that \(y\ne z\).
Next we note that some congruence and divisibility constraints come for free:
Lemma
 (i)
\(z\equiv \frac{4}{3}k(2d^2)+9(k+d)\pmod {18}\);
 (ii)
if \(p\equiv 2\pmod {3}\) then \(t\le 3s\);
 (iii)
if \(t\le 3s\) then \(s\equiv t\pmod {2}\);
 (iv)
if \({{\,\mathrm{ord}\,}}_p{k}\in \{1,2\}\) then \(s\in \{0,{{\,\mathrm{ord}\,}}_p{k}\}\).
Proof

If \(p=3\) then \(s=t=0\), so \(s\equiv t\pmod {2}\).

If \(p\ne 3\) and \(3s>t+2{{\,\mathrm{ord}\,}}_p{2}\) then \({{\,\mathrm{ord}\,}}_p\Delta =s+t+2{{\,\mathrm{ord}\,}}_p2\), so \(s\equiv t\pmod {2}\).

If \(3s\in \{t,t+2\}\) then \(s\equiv t\pmod {2}\).

If \(p=2\) and \(3s=t+1\) then \(2^{4s}\Delta =3\bigl (2uvu^4\bigr )\equiv 3\pmod {4}\), which is impossible.
Thus, once the residue class of \(z\pmod {d}\) is fixed, its residue modulo \({{\,\mathrm{lcm}\,}}(d,18)\) is determined. Note also that conditions (ii) and (iii) are efficient to test for \(p=2\).
Thus, when B fits within the machine word size, we expect the running time to be nearly linear, and this is what we observe in practice for \(B< 2^{64}\).
3 Implementation
We implemented the above algorithm in C, with a few inline assembly routines for Montgomery arithmetic [14] written by Buhrow [4], and Walisch’s primesieve library [19] for enumerating prime numbers.
The algorithm is naturally split between values of d with a prime factor exceeding \(\sqrt{\alpha B}\) and those that are \(\sqrt{\alpha B}\)smooth. The former set of d consumes more than twothirds of the running time, but is more easily parallelized. We ran this part on the massively parallel cluster Bluecrystal Phase 3 at the Advanced Computing Research Centre, University of Bristol. For the smooth d we used a separate small cluster of 32 and 64core nodes.
The running time was approximately 8 coreyears per number tested.
Notes
Acknowledgements
I thank Roger HeathBrown for helpful comments and suggestions.
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