# Is it true that $\{x^3-2x+y^3-2y+z^3-2z: x,y,z\in\mathbb Z\}=\mathbb Z$?

A well known conjecture states that $$\{x^3+y^3+z^3:\ x,y,z\in\mathbb Z\}=\{m\in\mathbb Z:\ m\not\equiv\pm4\pmod 9\}.$$ For $$m=33,\, 42$$ an integer solution to the equation $$x^3+y^3+z^3=m$$ was only found last year.

In 2017, Tyrell asked whether $$\left\{\frac{x(x+1)(x+2)}6+\frac{y(y+1)(y+2)}6+\frac{z(z+1)(z+2)}6:\ x,y,z\in\mathbb Z\right\}=\mathbb Z,$$ see the question with the website http://math.stackexchange.com/questions/2472205. Few weeks ago Alkan (cf. Numbers of the form $x^2(x-1) + y^2(y-1) + z^2(z-1)$ with $x,y,z\in\mathbb Z$) conjectured that $$\left\{\frac{x^2(x-1)}2+\frac{y^2(y-1)}2+\frac{z^2(z-1)}2:\ x,y,z\in\mathbb Z\right\}=\mathbb Z.$$

I think it's interesting to find a cubic polynomial $$P(x)$$ with integer coefficients such that $$\{P(x)+P(y)+P(z):\ x,y,z\in\mathbb Z\}=\mathbb Z.$$ This led me to pose the following conjecture.

Conjecture. Each $$m\in\mathbb Z$$ can be written as a sum of three numbers of the form $$x^3-2x\ (x\in\mathbb Z)$$. In other words, we have $$\{x^3-2x+y^3-2y+z^3-2z: x,y,z\in\mathbb Z\}=\mathbb Z.$$

As $$P(x)=x^3-2x$$ is an odd function, the conjecture can be reduced to the case $$m\in\mathbb N=\{0,1,2,\ldots\}$$. Via computation I found that those natural numbers $$n\le1000$$ not in the set $$\{x^3-2x+y^3-2y+z^3-2z:\ x,y,z\in\{-1000,\ldots,1000\}\}$$ are $$\begin{gather}70,\ 75,\ 83,\ 86,\ 139,\ 185,\ 198,\ 237,\ 253,\ 262,\ 275, \ 305,\ 338,\ 355,\ 362, \\397, 414,\ 415,\ 422,\ 426,\ 457,\ 458,\ 509,\ 535,\ 558,\ 562,\ 564,\ 580,\ 583, \\ 593, \ 613,\ 614,\ 635,\ 642,\ 673,\ 677,\ 684, \ 693,\ 697,\ 722,\ 735,\ 779,\ 782, \\ 790,\ 791,\ 793,\ 807,\ 818,\ 850,\ 851,\ 870,\ 888,\ 898,\ 908,\ 943,\ 957. \end{gather}$$ Let $$S$$ denote the set of these numbers.

QUESTION. Can we find an explicit solution of the equation $$n=x^3-2x+y^3-2y+z^3-2z\ \ (x,y,z\in\mathbb Z)$$ for each $$n\in S$$?

• @Zhi-Wei-Sun: Solutions for $n=185,198,614,793$ were found. Commented Jul 27, 2020 at 10:37
• No solution for $338,422,426,509,558,583$ exists where one of $|x|,|y|,|z|$ does not exceed $6\times10^5$. Commented Jul 27, 2020 at 10:45
• Solutions for n=422,426,509,583 were found. Commented Jul 28, 2020 at 2:31
• $338= P(109043424)+ P(223729659)+ P(-232050701).$ Only $n=558$ remains. Commented Jul 29, 2020 at 1:54
• I could not find a solution for $n=558$ using modulus method with $|x|,|y|,|z|<10^{11}$. To verify this result, I'll use LLL method with PARI-GP, which takes about 20 days. Commented Aug 2, 2020 at 21:18

           0 = P(7) + P(10) + P(-11)
= P(3250) + P(2293) + P(-3593)
= P(6266) + P(13243) + P(-13695)
= P(11700) + P(13277) + P(-15797)
= P(37555) + P(131381) + P(-132396)
= P(747511) + P(1059490) + P(-1171307)
= P(5529835) + P(22681597) + P(-22790636)
= P(8042677) + P(13682243) + P(-14552100)
= P(14270088) + P(39054467) + P(-39679475)
= P(29292092) + P(81358953) + P(-82605425)
= P(42588445) + P(291524359) + P(-291827018)
= P(56973565) + P(71715599) + P(-82119294)
= P(35977605) + P(866776048) + P(-866796709)
= P(143141833) + P(102053460) + P(-158684449)
= P(784428376) + P(3091918585) + P(-3108657737)
= P(129810373) + P(136917575) + P(-168147294)

Though I expanded the search range to 10^10, no solution for n=558 was found.
On the other hand, there are many solutions for n=1 below.

1 = P(1439) + P(2554) + P(-2698)
= P(-1506) + P(-2432) + P(2611)
= P(-5214) + P(-11006) + P(11383)
= P(-8516) + P(-17400) + P(18055)
= P(13952) + P(70243) + P(-70426)
= P(18457) + P(10233) + P(-19451)
= P(18949) + P(56163) + P(-56873)
= P(21394) + P(107636) + P(-107917)
= P(21599) + P(61917) + P(-62781)
= P(75215) + P(256620) + P(-258756)
= P(132479) + P(517316) + P(-520196)
= P(525599) + P(2589115) + P(-2596315)
= P(697638) + P(803074) + P(-950033)
= P(-140064) + P(-198656) + P(219583)
= P(-198846) + P(-913333) + P(916464)
= P(-257810) + P(-1509380) + P(1511883)
= P(-617569) + P(-1930917) + P(1951749)
= P(-887510) + P(-1092290) + P(1260399)
= P(-931224) + P(-1288696) + P(1433823)
= P(1384739) + P(2458622) + P(-2597096)
= P(1602719) + P(9519294) + P(-9534414)
= P(4092479) + P(28437689) + P(-28465913)
= P(4875121) + P(2381859) + P(-5057717)
= P(9192959) + P(73135432) + P(-73183816)
= P(-1288696) + P(-931224) + P(1433823)
= P(-6063625) + P(-20241211) + P(20420995)
= P(-6919820) + P(-21816096) + P(22045735)
= P(-8121991) + P(-32025689) + P(32198879)
= P(18740159) + P(167927031) + P(-168004791)
= P(24544311) + P(124666228) + P(-124982552)
= P(62900639) + P(689911189) + P(-690085429)
= P(96931304) + P(198453683) + P(-205880474)
= P(-11745176) + P(-17900062) + P(19447931)
= P(-20241211) + P(-6063625) + P(20420995)
= P(-24301082) + P(-68349676) + P(69358667)
= P(-41154429) + P(-47640292) + P(56234034)
= P(-42083576) + P(-117387233) + P(119163144)
= P(-95843081) + P(-181052723) + P(189595899)
= P(106254719) + P(1271978124) + P(-1272225228)
= P(123437629) + P(177749151) + P(-195715037)
= P(-119444557) + P(-275690964) + P(282970698)
= P(-120282709) + P(-113449262) + P(147367664)
= P(-169017105) + P(-182314167) + P(221641415)
= P(-181052723) + P(-95843081) + P(189595899)
= P(-190571214) + P(-1169296181) + P(1170981088)
= P(-1129360025) + P(-3749403040) + P(3783251348)

Solution for n=338 was found using LLL algorithm for X^3+Y^3=1.
338= P(109043424)+ P(223729659)+ P(-232050701)
Only n=558 remains.

422= P(31441077)+ P(52488141)+ P(-56007428)
426= P(-11575473)+ P(-42374626)+ P(42660619)
509= P(4620839)+ P(7911642)+ P(-8405584)
583= P(-2697799)+ P(-3187069)+ P(3732685)

185=P(-14114372)+ P(-283189)+ P(14114410)
198=P(-142960)+ P(-613349)+ P(615927)
614=P(-412307)+ P(-16619)+ P(412316)
793=P(-296708)+ P(-387970)+ P(438851)

262 = P(10239) + P(5400) + P(-10717)
275 = P(38314) + P(4857) + P(-38340)
305 = P(8535) + P(5187) + P(-9131)
355 = P(2568) + P(982) + P(-2615)
362 = P(6547) + P(636) + P(-6549)
397 = P(-2029) + P(-973) + P(2101)
414 = P(1059) + P(576) + P(-1113)
457 = P(-7709) + P(-6134) + P(8832)
535 = P(-11999) + P(-2241) + P(12025)
562 = P(-3435) + P(-862) + P(3453)
564 = P(-848) + P(-751) + P(1011)
580 = P(-2295) + P(-825) + P(2330)
593 = P(1563) + P(458) + P(-1576)
613 = P(18873) + P(1623) + P(-18877)
635 = P(10566) + P(9745) + P(-12816)
642 = P(-5020) + P(-3871) + P(5693)
673 = P(4487) + P(566) + P(-4490)
677 = P(5967) + P(1087) + P(-5979)
684 = P(4316) + P(2750) + P(-4660)
693 = P(3575) + P(702) + P(-3584)
697 = P(-17181) + P(-2952) + P(17210)
722 = P(-1051) + P(-311) + P(1060)
735 = P(1934) + P(1460) + P(-2179)
779 = P(3781) + P(1593) + P(-3873)
790 = P(-152491) + P(-8563) + P(152500)
791 = P(11265) + P(8599) + P(-12735)
818 = P(2003) + P(874) + P(-2057)
850 = P(9047) + P(1510) + P(-9061)
851 = P(1105) + P(264) + P(-1110)
870 = P(6390) + P(1917) + P(-6447)
888 = P(3928) + P(1444) + P(-3992)
898 = P(1709) + P(929) + P(-1796)
908 = P(4950) + P(4172) + P(-5788)
943 = P(-5848) + P(-3743) + P(6320)
957 = P(-4297) + P(-3091) + P(4775)


Let $$P(x):=x^3-2x$$. Then $$\begin{gather} 70=P(2714)+P(1367)+P(-2825),\\ 75=P(16333)+P(14200)+P(-19328),\\ 83=P(6714)+P(-6682)+P(-1627),\\ 86=P(6413)+P(3721)+P(-6806). \end{gather}$$

• Dr. Deyi Chen has informed me that he has found $139=P(-105811)+P(105801)+P(6951)$. Commented Jul 27, 2020 at 5:46
• More contributions from Dr. Deyi Chen: \begin{align}237=&P(-54523)+P(54267)+P(13147),\\262=&P(-2719)+P(2712)+P(537),\\275=&P(-38340)+P(38314)+P(4857),\\305=&P(-9131)+P(8535)+P(5187),\\355=&P(-2615)+P(2568)+P(982).\end{align} Commented Jul 27, 2020 at 8:10
• Further contributions from Dr. Deyi Chen: \begin{align}253=&P(-63060)+P(61067)+P(28452),\\415=&P(-80702)+P(79605)+P(27652),\\458=&P(-32329)+P(-12710)+P(32971),\\782=&P(-74024)+P(-56403)+P(83637),\\807=&P(-68697)+P(62165)+P(43789).\end{align} Commented Jul 27, 2020 at 9:03
• This seems inappropriate as an answer; it would be a comment to the question or to the more highly-voted and more informative answer.
– user44143
Commented Jul 31, 2020 at 17:32
• @Matt an upvote merely means "this answer is useful". Commented Aug 1, 2020 at 11:46