Dependencies

Let $a,b,c\in {\mathcal{O}}_K$ such that $c\ne 0$ and $gcd(a,b)=1$.
Let $\lambda \nmid a$, $\lambda \nmid b$, $\lambda \mid c$ and ${\lambda }^2\mid a+b$.

A solution is a tuple $S=(a,b,c,u)$ satisfying the equation $a^3+b^3=u{c}^3$.

Let $a,b,c\in {\mathcal{O}}_K$ such that $c\ne 0$ and $gcd(a,b)=1$.
Let $\lambda \nmid a$, $\lambda \nmid b$ and $\lambda \mid c$.

A solution’ is a tuple $S^{\prime }=(a,b,c,u)$ satisfying the equation $a^3+b^3=u{c}^3.$

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$ with multiplicity $n$.
Let $S_f^{\prime }=(Y,u_{4}Z,{\lambda }^{n-1}X,u_5)$.

Then $S_f^{\prime }$ is a $solutio{n}^{\prime }$.

Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

The multiplicity of $S^{\prime }$ is the largest $n\in \mathbb{N}$ such that ${\lambda }^n\mid c$.

Let $S=(a,b,c,u)$ be a $solution$.

We say that $S$ is minimal if for all solutions $S_1=(a_1,b_1,c_1,u_1)$, the multiplicity of $S$ is less than or equal to the $multiplicity$ of $S_1$.

Let $S=(a,b,c,u)$ be a $solution$.

The multiplicity of $S$ is the largest $n\in \mathbb{N}$ such that ${\lambda }^n\mid c$.

Let $S$ be a $solution$.

We define $u_1\in {\mathcal{O}}_K^{\times }$ and $X\in {\mathcal{O}}_K$ such that $x=u_1{X}^3$.
We define $u_2\in {\mathcal{O}}_K^{\times }$ and $Y\in {\mathcal{O}}_K$ such that $y=u_2{Y}^3$.
We define $u_3\in {\mathcal{O}}_K^{\times }$ and $Z\in {\mathcal{O}}_K$ such that $z=u_3{Z}^3$.
We define $u_4=\eta u_3{u}_2^{-1}$.
We define $u_5=-{\eta }^2{u}_1{u}_2^{-1}$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.

We define $x\in {\mathcal{O}}_K$ such that $a+b={\lambda }^{3n-2}x$.
We define $y\in {\mathcal{O}}_K$ such that $a+\eta b=\lambda y$.
We define $z\in {\mathcal{O}}_K$ such that $a+{\eta }^{2}b=\lambda z$.
We define $w\in {\mathcal{O}}_K$ such that $c={\lambda }^{n}w$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $S=(a,b,c,u)$ be a $solution$.

Then $a+\eta b=(a+b)+\lambda b$.

Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then ${\lambda }^4\mid a^3-1\wedge {\lambda }^4\mid b^3+1$ or ${\lambda }^4\mid a^3+1\wedge {\lambda }^4\mid b^3-1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.
Let $p\in {\mathcal{O}}_K$ be a prime such that $p\mid a+b$ and $p\mid a+\eta b$.

Then $p$ is associated with $\lambda$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.
Let $p\in {\mathcal{O}}_K$ be a prime such that $p\mid a+b$ and $p\mid a+{\eta }^{2}b$.

Then $p$ is associated with $\lambda$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.
Let $p\in {\mathcal{O}}_K$ be a prime such that $p\mid a+\eta b$ and $p\mid a+{\eta }^{2}b$.

Then $p$ is associated with $\lambda$.

Let $S$ be a $solution$.

Then $u_4\in \{-1,1\}\subset {\mathcal{O}}_K$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $I$ be the ideal generated by $\lambda$.

Then ${\mathcal{O}}_K/I$ has cardinality $3$.

Let $S$ be a $solution$ with multiplicity $n$.

Then $gcd(x,y)=1$.

Let $S$ be a $solution$.

Then $gcd(x,z)=1$.

Let $S$ be a $solution$.

Then $gcd(Y,Z)=1$.

Let $S$ be a $solution$.

Then $gcd(y,z)=1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then $a^3+b^3=(a+b)(a+\eta b)(a+{\eta }^{2}b)$.

Let ${\mathbb{Z}}_9$ be the ring of integers modulo $9$.
Let ${\mathbb{Z}}_3$ be the ring of integers modulo $3$.
Let $n\in {\mathbb{Z}}_9$.
Let $\varphi :{\mathbb{Z}}_9\to {\mathbb{Z}}_3$ be the canonical ring homomorphism.
Let $\varphi (n)\ne 0$.

Then $n^3=1\vee n^3=8$.

Let $n\in \mathbb{N}$.
Let $\left[n\right]\in {\mathbb{Z}}_9$.
Let $3\nmid n$.

Then ${\left[n\right]}^3=1\vee {\left[n\right]}^3=8$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $x\in {\mathcal{O}}_K$.

Then $x^3-1=(x-1)(x-\eta )(x-{\eta }^2)$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $x\in {\mathcal{O}}_K$.

Then $(\lambda \mid x)\vee (\lambda \mid x-1)\vee (\lambda \mid x+1)$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $u\in {\mathcal{O}}_K^{\times }$ be a unit.

If $\mathrm{\exists }m\in \mathbb{Z}$ such that ${\lambda }^2\mid u-m$, then $u=1\vee u=-1$.
This is a special case of the Kummer’s Lemma.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $S=(a,b,c,u)$ be a $solution$.

Then $(\eta +1)(-\eta )=1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.

Then $\eta$ is a unit.

Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then $\mathrm{\exists }{a}_1,b_1\in {\mathcal{O}}_k$ such that $S_1=(a_1,b_1,c,u)$ is a $solution$.

Let $S$ be a $solution$ with multiplicity $n$.

Then there is a minimal solution $S_1$.

Let $S$ be a $solution$ with multiplicity $n$.

Then there is a $solution$ with multiplicity $m<n$.

Let $S^{\prime }$ be a $solutio{n}^{\prime }$ with multiplicity $n$.

Then there is a $solutionS$ with multiplicity $n$.

To prove Theorem 3.66, it suffices to prove Theorem 3.64.
Equivalently, Theorem 3.64 implies Theorem 3.66.

Let $R$ be a commutative semiring, domain and normalised $gcd$ monoid.
Let $a,b,c\in R$.
Let $n\in \mathbb{N}$.

Then, to prove Fermat’s Last Theorem for exponent $n$ in $R$, one can assume, without loss of generality, that $gcd(a,b,c)=1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$ with multiplicity $n$.

Then $Y^3+(u_{4}Z{)}^3=u_5({\lambda }^{n-1}X{)}^3$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$ with multiplicity $n$.

Then $u_1{X}^3{\lambda }^{3n-2}+u_2\eta Y^3\lambda +u_3{\eta }^2{Z}^3\lambda =0$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$ with multiplicity $n$.

Then $Y^3+u_4{Z}^3=u_5({\lambda }^{(}n-1)X{)}^3$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.

Then $\lambda \mid a+\eta b$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$.

Then $\lambda \mid a+{\eta }^{2}b$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $x\in {\mathcal{O}}_K$.

Then $\lambda \mid x(x-1)(x-(\eta +1))$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then $\lambda \mid 3$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then $\lambda \ne 0$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then $\lambda \nmid 2$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid w$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid X$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid x$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid Y$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid y$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid Z$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then $\lambda \nmid z$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then $\lambda$ is not a unit.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S=(a,b,c,u)$ be a $solution$ with multiplicity $n$.

Then ${\lambda }^{3n-2}\mid a+b$.

Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then ${\lambda }^4\mid c^3$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $x\in {\mathcal{O}}_K$.

If $\lambda \mid x+1$, then ${\lambda }^4\mid x^3+1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $x\in {\mathcal{O}}_K$.

If $\lambda \mid x-1$, then ${\lambda }^4\mid x^3-1$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $x\in {\mathcal{O}}_K$.

If $\lambda \nmid x$, then $({\lambda }^4\mid x^3-1)\vee ({\lambda }^4\mid x^3+1)$.

Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then ${\lambda }^2\mid c$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then $\lambda$ is prime.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.

Then ${\lambda }^2=-3\eta$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$.

Then ${\lambda }^2\mid {\lambda }^4$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S$ be a $solution$ with multiplicity $n$.

Then ${\lambda }^2\mid u_5({\lambda }^{n-1}X{)}^3$.

Let $K=\mathbb{Q}({\zeta }_3)$ be the third cyclotomic field.
Let ${\mathcal{O}}_K=\mathbb{Z}[{\zeta }_3]$ be the ring of integers of $K$.
Let ${\mathcal{O}}_K^{\times }$ be the group of units of ${\mathcal{O}}_K$.
Let ${\zeta }_3\in K$ be any primitive third root of unity.
Let $\eta \in {\mathcal{O}}_K$ be the element corresponding to ${\zeta }_3\in K$.
Let $\lambda \in {\mathcal{O}}_K$ be such that $\lambda =\eta -1$.
Let $S^{\prime }=(a,b,c,u)$ be a $solutio{n}^{\prime }$.

Then $({\lambda }^2\mid a+b)\vee ({\lambda }^2\mid a+\eta b)\vee ({\lambda }^2\mid a+{\eta }^{2}b)$.