[proofplan] We prove the result by infinite descent. After reducing to a primitive solution, we work in the Eisenstein integer ring $\mathbb{Z}[\omega]$, where unique factorization controls the factors in $x^3+y^3=(x+y)(x+\omega y)(x+\omega^2y)$. We verify the coprimality hypotheses needed for the Eisenstein descent theorem, then cite that theorem to construct a strictly smaller primitive positive solution, contradicting the [well-ordering principle](/theorems/721) for positive integers. [/proofplan]

[step:Reduce a putative solution to a primitive one] Assume, for contradiction, that positive integers $x,y,z \in \mathbb{Z}$ satisfy \begin{align*} x^3+y^3=z^3. \end{align*} Let $d:=\gcd(x,y,z)$. If $d>1$, then $d^3$ divides each term and the triple $(x/d,y/d,z/d)$ is again a positive integer solution. Repeating this finite division process gives a positive integer solution $(a,b,c) \in \mathbb{Z}^3$ with \begin{align*} a^3+b^3=c^3, \qquad \gcd(a,b,c)=1. \end{align*} Among all primitive positive solutions choose one with $c$ minimal; this is allowed by the well-ordering principle for nonempty subsets of $\mathbb{N}$. [/step]

[step:Factor the equation in the Eisenstein integers] Let \begin{align*} \omega:=\frac{-1+\sqrt{-3}}{2}\in \mathbb{C} \end{align*} be a primitive cube root of unity, so $\omega^2+\omega+1=0$ and $\omega^3=1$. Define the Eisenstein integer ring \begin{align*} \mathbb{Z}[\omega]:=\{m+n\omega:m,n\in\mathbb{Z}\}\subset \mathbb{C}. \end{align*} The norm map $N:\mathbb{Z}[\omega]\to\mathbb{Z}_{\ge 0}$ is \begin{align*} N(m+n\omega):=(m+n\omega)(m+n\omega^2)=m^2-mn+n^2. \end{align*} By the [Euclidean property of the Eisenstein integers](/page/Eisenstein%20Integer), $\mathbb{Z}[\omega]$ is a unique factorization domain. In $\mathbb{Z}[\omega]$ we have \begin{align*} a^3+b^3=(a+b)(a+\omega b)(a+\omega^2b)=c^3. \end{align*} [/step]

[step:Verify the coprimality hypotheses for the Eisenstein descent theorem] We first show that the primitive condition implies pairwise coprimality. If a prime number $p\in\mathbb{Z}$ divides both $a$ and $b$, then $p^3$ divides $a^3+b^3=c^3$, hence $p$ divides $c$, contradicting $\gcd(a,b,c)=1$. If $p$ divides both $a$ and $c$, then $p^3$ divides $c^3-a^3=b^3$, so $p$ divides $b$, again a contradiction. The case where $p$ divides both $b$ and $c$ is identical with $a$ and $b$ interchanged. Hence \begin{align*} \gcd(a,b)=\gcd(a,c)=\gcd(b,c)=1. \end{align*} Because $a$ and $b$ are positive integers, $ab\ne 0$. [/step]

[step:Apply the cited Eisenstein descent theorem to obtain a smaller primitive solution] We use the [Eisenstein descent theorem for the cubic Fermat equation](/page/Eisenstein%20Integer): if $r,s,t\in\mathbb{Z}$ are pairwise coprime, $rs\ne 0$, and \begin{align*} r^3+s^3=t^3, \end{align*} then there exist positive integers $r_1,s_1,t_1\in\mathbb{Z}$ satisfying \begin{align*} r_1^3+s_1^3=t_1^3,\qquad \gcd(r_1,s_1,t_1)=1,\qquad 0<t_1<|t|. \end{align*} This theorem is proved in $\mathbb{Z}[\omega]$ using the Euclidean property, the classification of units, the ramified prime $1-\omega$ above $3$, and the resulting gcd calculation for the three Eisenstein factors. The previous step verifies its hypotheses for $(r,s,t)=(a,b,c)$, and $c>0$, so the theorem gives positive integers $a_1,b_1,c_1\in\mathbb{Z}$ such that \begin{align*} a_1^3+b_1^3=c_1^3,\qquad \gcd(a_1,b_1,c_1)=1,\qquad 0<c_1<c. \end{align*} [/step]

[step:Contradict minimality and conclude nonexistence] The existence of $(a_1,b_1,c_1)$ contradicts the choice of $(a,b,c)$ as a primitive positive solution with minimal third component $c$. Therefore no primitive positive solution exists. Since every positive solution reduces to a primitive positive solution by dividing by the common greatest divisor, no positive integers $x,y,z\in\mathbb{Z}$ satisfy $x^3+y^3=z^3$. [/step]