# Large initial solutions to $x^3+y^3 = Nz^3$?

Let $x,y,z$ be non-zero integers. Is it true that the initial or smallest solution (in terms of absolute value) to,

$$x^3+y^3 = Nz^3\tag1$$

for $N=94$ is,

$$15642626656646177^3 + (-15616184186396177)^3 = 94\cdot 590736058375050^3\,?$$

If not, then what is the largest initial solution for $N<100$? Or $N<200$?

P.S. Related posts are $x^3+y^3 = 6z^3$, and $x^3+y^3 = 22z^3$, and $x^3+y^3 = 313^2z^3$. See also this paper by Dasgupta and Voight for more details (including the elliptic curve for eq.1).

• This is the value given at oeis.org/A190356 but there's a warning there that it may not be the smallest solution for $N=94$. Commented Mar 18, 2015 at 6:42
• See also Table 1.4 of Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories by Yu. I. Manin and Alexei A. Panchishkin. Commented Mar 18, 2015 at 6:44
• But what you really want to see is Andrew Bremner, Positively prodigious powers or how Dudeney done it?, Math Mag 84 (2011) 120-125. Commented Mar 18, 2015 at 6:52
• @GerryMyerson: Thanks! That OEIS link referred to Hisanori Mishima's list with $N<1000$. The paper by Dasgupta does refer to the solution for $N=94$ as a (Mordell-Weil) generator (but of course there could be a smaller generator). Commented Mar 18, 2015 at 7:40
• Can anyone help me find (with a computer search) nonzero integers $a,b,c,d,e,n$ such that $$a^3-nb^3=c^3-nd^3=e^3$$ where $(a,b,e)$ and $(c,d,e)$ are pairwise coprime and $n^2\ne 1$ One solution set will do. Thanks. Commented Mar 26, 2021 at 2:19

After some insight courtesy of Achille Hui, I was able to answer my own question. It is well known (see also this) that $x^3+y^3=N$ is birationally equivalent to the elliptic curve $u^3-432N^2=v^2$ using the transformation $x=\frac{36N+v}{6u}$, $y=\frac{36N-v}{6u}$.

In this comment, Hui suggested the command,

Q$<x>$ := PolynomialRing(Rationals()); E00 := EllipticCurve(x^3-432*94^2); Q00 := Generators(E00); Q00;

on the Magma online calculator. We then find,

(62511752209/2480625 : -15629405421521177/3906984375 : 1)

Substituting this onto the transformation, we get,

$$x = \frac{15642626656646177}{590736058375050}\\ y = \frac{-15616184186396177}{590736058375050}$$

thus Magma confirms the solution given in the original question is indeed the smallest.

P.S. Incidentally, the numerators nicely factor as,

$$15642626656646177-15616184186396177 =2^4\cdot 3^8\cdot5^6 \cdot 7^3\cdot47$$

• The PARI/GP equivalent of the Magma code is Eoo = ellinit([0, -432*94^2]); Qoo = ellgenerators(Eoo). Commented Apr 27 at 2:45