Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:02:20Z http://mathoverflow.net/feeds/question/58188 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58188/are-nontrivial-integer-solutions-known-for-x3y3z33 Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? András Salamon 2011-03-11T19:05:47Z 2011-06-02T18:48:24Z <p>The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions known at that time.</p> <blockquote> <p>Are any other solutions known?</p> </blockquote> <p>By a conjecture of Tyszka, it would follow that if this equation had finitely many roots, then each component of a solution tuple would be at most $2^{2^{12}/3} \lt 2^{1365.34}$ in absolute value. (To see this, it is enough to express the equation using a Diophantine system in 13 variables in the form considered by Tyszka.) This leaves a large gap, since Elsenhans and Jahnel only considered solutions with components up to $10^{14} \approx 2^{46.5}$ in absolute value. It is also not obvious whether Tyszka's conjecture is true.</p> <p>OEIS sequence <a href="http://oeis.org/A173515" rel="nofollow">A173515</a> refers to equations of the form $x^3+y^3=z^3-n$, for $n$ a positive integer, as "Fermat near-misses". Infinite families of solutions are known for $n=\pm 1$, including one constructed by Ramanujan from generating functions (see Rowland's survey).</p> <ul> <li>Andreas-Stephan Elsenhans and Jörg Jahnel, <em>New sums of three cubes</em>, Math. Comp. <strong>78</strong> (2009), 1227–1230. DOI: <a href="http://dx.doi.org/10.1090/S0025-5718-08-02168-6" rel="nofollow">10.1090/S0025-5718-08-02168-6</a>. (<a href="http://www.staff.uni-bayreuth.de/~btm216/elk_ants6c.pdf" rel="nofollow">preprint</a>)</li> <li>Apoloniusz Tyszka, <em>A conjecture on integer arithmetic</em>, Newsletter of the European Mathematical Society (75), March 2010, 56–57. (<a href="http://www.ems-ph.org/journals/newsletter/pdf/2010-03-75.pdf" rel="nofollow">issue</a>)</li> <li>Eric S. Rowland, <em>Known Families of Integer Solutions of $x^3+y^3+z^3=n$</em>, 2005. (<a href="http://www.math.tulane.edu/~erowland/papers/koyama.pdf" rel="nofollow">manuscript</a>)</li> </ul> http://mathoverflow.net/questions/58188/are-nontrivial-integer-solutions-known-for-x3y3z33/58192#58192 Answer by Luis H Gallardo for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? Luis H Gallardo 2011-03-11T19:23:22Z 2011-03-11T19:23:22Z <p>Of course the problem is old and probably there is no hope to be resolved.</p> <p>The following papers of Vaserstein are of interest:</p> <p>MR1196532 (93k:11090) Payne, G.(1-PAS); Vaserstein, L.(1-PAS) Sums of three cubes. The arithmetic of function fields (Columbus, OH, 1991), 443–454, Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin, 1992. 11P05</p> <p>MR1284068 (95g:11128) Conn, W.(1-PAS); Vaserstein, L. N.(1-PAS) On sums of three integral cubes. (English summary) The Rademacher legacy to mathematics (University Park, PA, 1992), 285–294, Contemp. Math., 166, Amer. Math. Soc., Providence, RI, 1994. 11Y50 (11D25) PDF Clipboard Series Chapter Make Link</p> <p>Over the last 40 years there have been various computational efforts to search for integer solutions to the equation $x^3+y^3+z^3 = t$ for small integers $t$. This paper describes a search that found solutions for $t = 39$ and $t = 84$, as well as a number of other solutions </p> <p>for small $t$ that are of interest for various reasons. The authors used a symbolic computation package on workstations, and used different search techniques for different regions of interest. They argue that their data supports the conjecture that solutions should exist for all $t$ satisfying the easy necessary condition that $t$ not be congruent to $\pm 4$ modulo 9; the only such $t$ less than 100 for which no solutions are known are now $30,33,42,52,74,75$. The algorithms of this paper are tuned to providing solutions for an interval of possible $t$, whereas a recent algorithm due to Heath-Brown is faster for a fixed value of $t$, although it requires significant precomputation whose complexity depends on the class number of ${\bf Q}(\root 3 \of t)$. An implementation of that algorithm by D. R. Heath-Brown, W. M. Lioen and H. J. J. te Riele [Math. Comp. 61 (1993), no. 203, 235--244; MR1202610 (94f:11132)] also discovered some of the solutions found in the article under review. </p> http://mathoverflow.net/questions/58188/are-nontrivial-integer-solutions-known-for-x3y3z33/66740#66740 Answer by Noam D. Elkies for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? Noam D. Elkies 2011-06-02T14:31:54Z 2011-06-02T18:48:24Z <p>Just noticed this question. I agree with L.H.Gallardo that the problem is old (see e.g. Problem D5 in UPINT = <em>Unsolved Problems in Number Theory</em> by R.K.Guy), but not that it is hopeless: the usual heuristics suggest that the number of solutions with $\max(|x|,|y|,|z|) \leq H$ should be asymptotic to a multiple of $\log H$, so further solutions should eventually emerge (though it may indeed be hopeless to prove anything close to the $\log H$ heuristic).</p> <p>See also my article</p> <p>Rational points near curves and small nonzero $|x^3-y^2|$ via lattice reduction, <em>Lecture Notes in Computer Science</em> <strong>1838</strong> (proceedings of ANTS-4, 2000; W.Bosma, ed.), 33-63 = <a href="http://arXiv.org/abs/math/0005139" rel="nofollow">math.NT/0005139</a> on the arXiv.</p> <p>Among other things it gives an algorithm for finding all solutions of $|x^3 + y^3 + z^3| \ll H$ with $\max(|x|,|y|,|z|) \leq H$ that should run (and in practice does run) in time $\widetilde{O}(H)$; since we expect the number of solutions to be asymptotically proportional to $H$, this means we find the solutions in little more time than it takes to write them down.</p> <p>D.J.Bernstein has implemented the algorithm efficiently, and reports on the results of his and others' extensive computations at <a href="http://cr.yp.to/threecubes.html" rel="nofollow">http://cr.yp.to/threecubes.html</a> .</p> <p><strong>EDIT:</strong> for the specific problem $x^3+y^3+z^3=3$, Cassels showed that any solution must satisfy $x\equiv y\equiv z \bmod 9$ in this brief article:</p> <p>A Note on the Diophantine Equation $x^3+y^3+z^3=3$, <em>Math. of Computation</em> <strong>44</strong> #169 (Jan.1985), 265-266.</p> <p>This uses cubic reciprocity, and is stronger than what one can obtain from congruence conditions. See also Heath-Brown's paper "The Density of Zeros of Forms for which Weak Approximation Fails" (<em>Math. of Computation</em> <strong>59</strong> #200 (Oct.1992), 613-623), where he gives corresponding conditions for the homogeneous equation $x^3 + y^3 + z^3 = 3w^3$ and also $x^3 + y^3 + z^3 = 2w^3$, and reports that</p> <p><em>In a letter to the author, Professor Colliot-Thélène has shown that the above congruence restrictions are exactly those implied by the Brauer-Manin obstruction. Moreover, for the general equation $x^3 + y^3 + z^3 = kw^3$, with a noncube integer $k$, there is always a nontrivial obstruction, eliminating two-thirds of the adèlic points.</em></p> http://mathoverflow.net/questions/58188/are-nontrivial-integer-solutions-known-for-x3y3z33/66760#66760 Answer by John R Ramsden for Are nontrivial integer solutions known for $x^3+y^3+z^3=3$? John R Ramsden 2011-06-02T18:09:10Z 2011-06-02T18:09:10Z <p>As $x^3 + y^3 = c$ for any given (suitable) c is an elliptic curve, perhaps a reasonable strategy would be to try various integers $f$, $g$ for which $c := f^3 - 3 g^3$ is small and establish the Mordell-Weil rank of the curve.</p> <p>If this is ever positive (for values other than the known solutions the OP mentioned) then one would establish that there were other non-trivial rational solutions, even if these had still not been found.</p> <p>Edit: Rereading the OP's post, I notice they are asking for <em>integer</em> solutions rather than rational solutions, and I recall now that there are rational parametrizations anyway. So perhaps this approach isn't very useful after all.</p>