When does $(a+b\sqrt n)^3+(a-b\sqrt n)^3=c^3$ have integer solutions $(a,b,c,n)$?

Question

From this post Where Fermat's Last Theorem fails, we find the nice,

$$(18+17\sqrt2)^3+(18-17\sqrt2)^3=42^3$$

Using this initial solution, an infinite more can be generated using P. Tait's identity,

$$\big(x(y^3 + z^3)\big)^3 + \big(y(-x^3 - z^3)\big)^3 = \big(z(x^3 - y^3)\big)^3$$

which is true if $x^3+y^3=z^3.\,$ For example, the first leads to a second,

$$(707472 + 276119 \sqrt{2})^3 + (707472 - 276119 \sqrt{2})^3 = 1106700^3$$

and so on infinitely. But it is also the case that,

$$(2+\sqrt{-2})^3+(2-\sqrt{-2})^3=(-2)^3$$ $$(121+23\sqrt{-11})^3+(121-23\sqrt{-11})^3=(-88)^3$$

all of which are unique factorization domains UFD ${\bf Q}(\sqrt n)$. So at first I thought it was peculiar to UFDs, but expanding,

$$(a+b\sqrt n)^3+(a-b\sqrt n)^3=c^3$$

we get the simpler,

$$2 a^3 + 6 a b^2 n = c^3\tag{eq.1}$$

After I tested various $n$, it didn't seem to be limited to UFDs. For example,

$$(256+11\sqrt{-41})^3+(256-11\sqrt{-41})^3=296^3$$

Question: Can $2 a^3 + 6 a b^2 n = c^3$ (labeled as eq.1) be turned into an elliptic curve similar to this post so we can easily find $n$ for which eq.1 is solvable? If not, what are the $-100 < n < 100$ such that $(1)$ has a solution? (Is it in the OEIS?)

P.S. We avoid square $n$, since by FLT $x^3+y^3=z^3$, it just yields trivial solutions $xyz = 0$. Or if $n=-3m^2$, then $(a,b,c) = (3m,\,1,\,0)$ so we avoid $abc =0$ as well.

In your Question, what do you mean by 'find $n$'? Do you mean to find $n$ for which $x^3+y^3=z^3$ has a solution in $\Bbb{Q}(\sqrt{n})$? Or do you require that $z\in\Bbb{Z}$? — Servaes, Jan 06 '24 at 23:00
@Servaes Yes, but for now we focus on the form $(a+b\sqrt{n})^3+(a-b\sqrt{n})^3 = c^3$ where c is a non-zero integer. P.S. The more general form of course is $(a+b\sqrt{n})^3+(c+d\sqrt{n})^3 = (e+f\sqrt{n})^3$ where $(a,b,c,d,e,f)$ are all non-zero, but I haven't found an example of that yet. — Tito Piezas III, Jan 07 '24 at 02:08
@Tito Piezas III,
(a+bm)^k+(a-bm)^k = (c+dm)^k+(c-dm)^k, for k=3 has solution, (a,b,c,d)=(2,3,5,1) & m=(3)^(1/2) and for k=5, a=(1280)^(1/2), b=(255)^(1/2),c=(5)^(1/2),d=(4085)^(1/2) & m=1 — David, Jan 12 '24 at 15:30
@David The equation $a^k+b^k = c^k+d^k$ can be solved in quadratic integers up to $k=11$ with the FLT-like restriction $abcd \neq 0$. With that restriction, it is unknown if it can be brought higher for $k > 11$. See this post for more details. — Tito Piezas III, Jan 13 '24 at 02:49
@Tito Piezas III. The reason I suggested the form, (a^k+b^k)=(c^k+d^k), is because I noticed in your comment that you are looking for a solution to equation,[(a+sqrt{bn})^3+(c+sqrt{dn})^3 = (e+sqrt{fn})^3]. This equation is asymetrical in the sence that the (LHS) has two terms & the (RHS) has only one term. Hence it could be difficult to find numerical solutions to it. Regarding degree thirteen for my proposed equation do read my comment for the (MSE) question with the link shown here. https://math.stackexchange.com/questions/4821037 — David, Jan 13 '24 at 19:31

Tomita · Accepted Answer · 2024-01-07T04:56:40.510

Elliptic curve version.

$2a^3+6ab^2n = c^3$ can be reduced to $v^2 = -12nx^4+6nx$ with $x=a/c,y=b/c, v=6nxy$.

Furthermore, we can tranform $v^2 = -12nx^4+6nx$ to $Y^2 = X^3-432n^3$ with $X=6n/x,Y=6nv/x^2$.

The solutions are derived from one of the generators, so it may not be the smallest.

           (n,a,b,c)
           (-197, 425716141676, 17561345519, -95739315770)
           (-195, 2028, 131, -2886)
           (-194, 306808280101845674, 10558172571357337, 261831462443345254)
           (-191, 19652, 821, -986)
           (-186, 162, 41, -666)
           (-185, 23715170572, 805056269, 21263774126)
           (-182, 152210180652894, 7057684092887, -107041044260514)
           (-179, 256, 53, -904)
           (-177, 1500, 5264447, -3534330)
           (-174, 6, 1, -18)
           (-173, 4000000, 1838360791, -2411975800)
           (-170, 144, 5, 132)
           (-167, 1, 1, -10)
           (-161, 4442376652, 236037857, -3994761778)
           (-159, 1029, 43, 714)
           (-158, 304423498, 2560955153, -12369700078)
           (-155, 25, 1, 20)
           (-149, 79433915648, 3751090247, 14738891176)
           (-146, 10658, 19847, -154322)
           (-143, 169, 5, 182)
           (-141, 607836, 6037, 755022)
           (-137, 5756278756, 1896400265, -25527545782)
           (-134, 2569424405062, 1495926864335, -16617711752314)
           (-131, 1, 107, -208)
           (-129, 12, 1, -18)
           (-123, 96, 5, -12)
           (-122, 7442, 389, -122)
           (-119, 49, 31, -322)
           (-114, 775235226, 52094663, -797509914)
           (-113, 63253004, 8638589, -139147874)
           (-110, 18, 1, -6)
           (-109, 4500, 253, -1830)
           (-107, 243, 29, -468)
           (-106, 18, 1, 6)
           (-105, 2888844, 141143, 2286942)
           (-101, 907924000, 52423201, -247766140)
           (-95, 361, 235, -2242)
           (-89, 36, 1, 42)
           (-87, 81, 5, 18)
           (-86, 16, 1, -4)
           (-83, 9, 5, -48)
           (-78, 1014, 58735, -117858)
           (-77, 67228, 83253539, -59932978)
           (-74, 407625226, 30897005, -334158322)
           (-71, 1, 5, -22)
           (-69, 15972, 1255, -13134)
           (-65, 676, 33581, -66742)
           (-62, 16, 5, -52)
           (-59, 9, 1, -12)
           (-53, 2304, 13609, -51384)
           (-51, 12, 1, -6)
           (-47, 4, 1, -10)
           (-42, 294, 2395, -7518)
           (-41, 256, 11, 296)
           (-38, 6498, 7057, -41838)
           (-35, 49, 5, -28)
           (-33, 12, 25, -114)
           (-31, 9, 1, -6)
           (-29, 288, 191, -1212)
           (-26, 3042, 307, 2262)
           (-23, 9, 1, 6)
           (-17, 4, 29, -70)
           (-15, 3, 1, -6)
           (-14, 98, 31, -182)
           (-11, 1, 1, -4)
           (-6, 6, 1, 6)
           (-5, 4, 1, 2)
           (-2, 2, 1, -2)
           (2, 18, 17, 42)
           (5, 9, 1, 12)
           (6, 6, 5, 18)
           (11, 246924, 168275, 789222)
           (14, 5058568998, 13722993145, 43138907994)
           (15, 12, 43, 126)
           (17, 36, 127, 390)
           (23, 377606781684, 274693407295, 1592635446402)
           (26, 10494278917902, 202621693481065, 406588108522206)
           (29, 243, 7295, 13104)
           (33, 12, 1, 18)
           (35, 284784377187481164, 154212287927062829, 1136627710868147862)
           (38, 2224710090824741532242298, 682526736456078049630579, 6368669335721179285647342)
           (41, 288, 40955, 49164)
           (42, 645918, 34007, 899262)
           (43, 4, 11, 50)
           (47, 71734416291045108, 2220116835128461, 94277936072724450)
           (51, 514500, 413221, 3005730)
           (53, 15166431, 10329073, 80493504)
           (58, 1682, 25, 2146)
           (59, 7358173787172289500, 8608159082661727764757, 5779158750471360655230)
           (62, 24617358, 1598549, 37619010)
           (65, 308172852, 30203753, 551979246)
           (69, 81, 55, 468)
           (71, 2203793333881819021004544, 9853319396397693506345, 2780540935484837082868632)
           (74, 5723994783817163598634351926, 1393441383372024857043537865, 17445840971557474688254138278)
           (77, 4261646277, 55593553, 5438792268)
           (78, 667722042, 298862351, 3054806118)
           (82, 2, 1, 10)
           (83, 101401546779607829751759181824, 78095516271323295246676139023, 676839307868352923167913503536)
           (85, 1, 1, 8)
           (86, 11054250, 7107523, 66255630)
           (87, 2670540192, 154541347, 4148286948)
           (89, 4500, 105999947, 30000030)
           (93, 3, 1, 12)
           (101, 569277036000, 6183969793, 725692439940)
           (105, 588, 565, 4914)
           (106, 128, 5, 184)
           (109, 4, 1, 14)
           (110, 2558788532262, 246175379371, 5140697537226)
           (113, 53757763452, 16747406609, 219208021578)
           (114, 2890815729966, 327923088175, 6390219852198)
           (119, 85778248500, 1543310322733, 5264153630610)
           (123, 38901504, 2108117, 62601336)
           (134, 69744445649988575971907142, 419714537773835540334275214965, 214564515347932269394392052626)
           (137, 4775436, 972389, 15780114)
           (141, 1500, 1, 1890)
           (142, 18, 1, 30)
           (149, 7716375, 414647, 12815880)
           (155, 18456894399119616, 25901779522726321, 225904518368817288)
           (158, 177521844998411170826779098, 1805092471425063874803121, 227259216648420795612185622)
           (159, 203504954086656, 133367294253005, 1513952552186568)
           (161, 8666532, 244085, 12165846)
           (170, 81927315839221632, 3202809636185995, 125083122754631496)
           (173, 14145645792229575872157, 705144424515109688777, 23490393278718337138440)
           (177, 324, 23, 630)
           (185, 493138113204901915743156, 38127741324790671795497, 1011726091067655550214238)
           (186, 105456, 1955, 140868)
           (191, 293286031033541406772830187500, 13454207475992182129546755850903, 39330626629787271278291036795550)
           (195, 34304062637364, 189340166991769, 1128964881768282)
           (197, 15991018092, 1748033461, 40398858570)

+1 Great! I knew there had to be an elliptic curve. How remarkable that the elliptic curve for $x^3+y^3 = N$ as in this post and for this post are so similar, $$u^3-432N^2 = v^2$$ $$X^3-432n^3 = Y^2$$ — Tito Piezas III, Jan 07 '24 at 05:06
Tomita, looking at your solutions, at first I thought $\pm n$ would both have solutions. However, since the elliptic curve is $X^3-2(6n)^3 = y^2$, then there is a qualitative difference between positive and negative cases. For example, I find n =+(43, 58, 82, 85, 93, 142) has no negative counterpart. Likewise, I find n = -(31, 95, 107, 122, 129, 131, 143, 146, 167, 174, 179, 182, 194) has no positive counterpart. Just for info. — Tito Piezas III, Jan 07 '24 at 05:51
For $n=(-19,-43,-67,-163)$, Magma says that elliptic curve has rank $0$ and has no integer points. — Tomita, Jan 07 '24 at 07:14
I have a non-monic quartic polynomial to be made a square $(ax^2+bx+c)(x+1)x = y^2$. What is the Magma command to solve this in the rationals where $x\neq-1$ and $x\neq0$? For example, $219(3x^2+3x+1)(x+1)x = y^2$. Any rational solution? — Tito Piezas III, Jan 19 '24 at 06:50
@Tito,First, transform quartic to elliptic curve $E.$ Next, use Magma command Generators($E$) and Rank($E$). $219(3x^2+3x+1)(x+1)x = y^2$ can be transformed to $Y^2=X^3+31974X -2723119.$ E := EllipticCurve([0, 0, 0, 31974, -2723119]); SetClassGroupBounds("GRH"); Rank(E); Generators(E);
Magma says $E$ has rank $0$ and has only solution $(X,Y)=(73,0).$ Please see here how to transform quartic to elliptic curve. — Tomita, Jan 19 '24 at 07:32
The linked post was actually edited by me. I always have trouble when the leading or ending coefficient is a non-square. So this is Case 2. In summary, I assume the only rational $x$ of $219(3x^2+3x+1)(x+1)x=y^2$ is $x = -1$ and $x = 0$? — Tito Piezas III, Jan 19 '24 at 07:45
Extrapolating from your cubic, I guess that the quartic $3d(3x^2+3x+1)(x+1)x = y^2$ can be transformed to the cubic $X^3+6d^2 X-7d^3 = Y^2$. I tested it on primes $d = p^2+3q^2$ and of primes $d\leq163$, only two failed, namely $d=73,97$. I thought they'd all pass. This has to do with $x^3+y^3+z^3 = t^3$. I'll post a new question soon. — Tito Piezas III, Jan 19 '24 at 08:23
@Tito, Yes, I think the quartic $3d(3x^2+3x+1)(x+1)x=y^2$ can be transformed to the cubic $X^3+6d^2X−7d^3=Y^2.$ — Tomita, Jan 19 '24 at 09:07

score 1 · Answer 2 · answered Jan 07 '24 at 02:07

I don't know this is the answer you want.

Search results for $2a^3+6ab^2n = c^3$ where $-100<n<100$ and $(a,b)<1000$.

$n$ is squarefree.

           (n,a,b,c)
           (-95, 361, 235, -2242)
           (-89, 36, 1, 42)
           (-87, 81, 5, 18)
           (-86, 16, 1, -4)
           (-83, 9, 5, -48)
           (-71, 1, 5, -22)
           (-62, 16, 5, -52)
           (-59, 9, 1, -12)
           (-51, 12, 1, -6)
           (-47, 4, 1, -10)
           (-41, 256, 11, 296)
           (-35, 49, 5, -28)
           (-33, 12, 25, -114)
           (-31, 9, 1, -6)
           (-29, 288, 191, -1212)
           (-23, 9, 1, 6)
           (-17, 4, 29, -70)
           (-15, 3, 1, -6)
           (-14, 98, 31, -182)
           (-11, 1, 1, -4)
           (-6, 6, 1, 6)
           (-5, 4, 1, 2)
           (-2, 2, 1, -2)
           (1, 1, 1, 2)
           (2, 18, 17, 42)
           (5, 9, 1, 12)
           (6, 6, 5, 18)
           (15, 12, 43, 126)
           (17, 36, 127, 390)
           (33, 12, 1, 18)
           (43, 4, 11, 50)
           (58, 16, 283, 764)
           (69, 81, 55, 468)
           (82, 2, 1, 10)
           (85, 1, 1, 8)
           (93, 3, 1, 12)

+1 Thanks! The sequence (2,5,6,11,14,15,17,23) has only one hit in OEIS but the next term is 26, and it doesn't seem to be related to $x^3+y^3 = z^3$. It is interesting that the $a$-variable are all of form $u^2,,2v^2,,3w^2$. Furthermore, abs(primes) of the form $n=-(6m-1)$ are represented, except $n=-53$. Can you extend the table to $-200<n<200$? (Converting $2a^3+6ab^2n=c^3$ to an elliptic curve would be ideal, since even if $n=-53$ has large solutions, then the online Magma calculator can find it.) — Tito Piezas III, Jan 07 '24 at 02:27
You may like this new post about elliptic curves and $a^4+b^4+c^4 =d^4$. — Tito Piezas III, Jan 28 '24 at 06:18

When does $(a+b\sqrt n)^3+(a-b\sqrt n)^3=c^3$ have integer solutions $(a,b,c,n)$?

2 Answers2

Linked