Dependencies

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

A solution’ is a tuple \(S'=(a, b, c, u)\) 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 \(\mathcal{O}^\times _K\) 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 \(S_f' = (Y,u_4 Z, \lambda ^{n-1} X, u_5)\).

Then \(S_f'\) is a \(solution'\).

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 \(S=(a, b, c, u)\) 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 \(u_1 \in \mathcal{O}^\times _K\) and \(X \in \mathcal{O}_K\) such that \(x = u_1 X^3\).
We define \(u_2 \in \mathcal{O}^\times _K\) and \(Y \in \mathcal{O}_K\) such that \(y = u_2 Y^3\).
We define \(u_3 \in \mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(S'=(a, b, c, u)\) be a \(solution'\).

Then \(\lambda ^4 \mid a^3 - 1 \land \lambda ^4 \mid b^3 + 1\) or \(\lambda ^4 \mid a^3 + 1 \land \lambda ^4 \mid b^3 - 1\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(u_4 \in \left\{ -1,1\right\} \subset \mathcal{O}_K\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(a^3 + b^3 = (a + b) (a + \eta b) (a + \eta ^2 b)\).

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 \(\phi : \mathbb {Z}_9 \to \mathbb {Z}_3\) be the canonical ring homomorphism.
Let \(\phi (n) \neq 0\).

Then \(n^3=1 \lor n^3=8\).

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

Then \(\left[n \right]^3 = 1 \lor \left[n \right]^3 = 8\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \((\lambda \mid x) \lor (\lambda \mid x-1) \lor (\lambda \mid x+1)\).

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 \(\mathcal{O}^\times _K\) 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 \(u \in \mathcal{O}^\times _K\) be a unit.

If \(\exists m \in \mathbb {Z}\) such that \(\lambda ^2 \mid u - m\), then \(u = 1 \lor u = -1\).
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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(S'=(a, b, c, u)\) be a \(solution'\).

Then \(\exists a_1,b_1 \in \mathcal{O}_k\) 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 \(m{\lt}n\).

Let \(S'\) be a \(solution'\) with multiplicity \(n\).

Then there is a \(solution\text{ }S\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\lambda \neq 0\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(S'=(a, b, c, u)\) be a \(solution'\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \((\lambda ^4 \mid x^3 - 1) \lor (\lambda ^4 \mid x^3 + 1)\).

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

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}^\times _K\) 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 \((\lambda ^2 \mid a + b) \lor (\lambda ^2 \mid a + \eta b) \lor (\lambda ^2 \mid a + \eta ^2 b)\).

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 \(\mathcal{O}^\times _K\) 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 \(\lambda ^2 \nmid a + \eta b\).

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 \(\mathcal{O}^\times _K\) 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 \(\lambda ^2 \nmid a + \eta ^2 b\).

Let \(S\) be a \(solution\) with multiplicity \(n\).

Then \(3n - 2 + 1 + 1 = 3n\).

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

Then the multiplicity of \(S'\) is finite.

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 \(\mathcal{O}^\times _K\) 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 the norm of \(\lambda \) is \(3\).

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 \(\mathcal{O}^\times _K\) 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 the norm of \(\lambda \) is a prime number.

Let \(S\) be a \(solution\) with multiplicity \(n\).

Then \(S_f'\) has multiplicity \(n-1\).

Let \(S\) be a \(solution\) with multiplicity \(n\).

Then \(S_f'\) has multiplicity \(m{\lt}n\).

Let \(S'=(a, b, c, u)\) be a \(solution'\) with multiplicity \(n\).

Then \(2 \leq n\).

Let \(S=(a, b, c, u)\) be a \(solution\) with multiplicity \(n\).

Then \(2 \leq n\).

Let \(a, b, c \in \mathbb {N}\).
Let \(3 \mid a\) and \(3 \mid b\).
Let \(a ^3 + b ^3 = c ^3\).

Then \(3 \mid \gcd (a,b,c)\).

Let \(a, b, c \in \mathbb {N}\).
Let \(3 \mid a\) and \(3 \mid c\).
Let \(a ^3 + b ^3 = c ^3\).

Then \(3 \mid \gcd (a,b,c)\).

Let \(a, b, c \in \mathbb {N}\).
Let \(3 \mid b\) and \(3 \mid c\).
Let \(a ^3 + b ^3 = c ^3\).

Then \(3 \mid \gcd (a,b,c)\).

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 \(\mathcal{O}^\times _K\) 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 \(\eta ^3 = 1\).

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 \(\mathcal{O}^\times _K\) 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 \(\eta ^2 + \eta + 1 = 0\).

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 \(\mathcal{O}^\times _K\) 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 \(\lambda \).
Let \(2 \in \mathcal{O}_K / I\).

Then \(2 \neq 0\).

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 \(\mathcal{O}^\times _K\) 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 \(\mathcal{O}_K / I = \left\{ 0, 1, -1\right\} \).

Let \(S\) be a \(solution\).

Then \(u_4\) is a unit.

Let \(S\) be a \(solution\).

Then \(u_5\) is a unit.

Let \(S\) be a \(solution\).

Then \(\exists u_1 \in \mathcal{O}^\times _K\) and \(\exists X \in \mathcal{O}_K\) such that \(x = u_1 X^3\).

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

Then \(x y z = u w^3\).

Let \(S\) be a \(solution\).

Then \(X \neq 0\).

Let \(S\) be a \(solution\).

Then \(\exists u_2 \in \mathcal{O}^\times _K\) and \(\exists Y \in \mathcal{O}_K\) such that \(y = u_2 Y^3\).

Let \(S\) be a \(solution\).

Then \(\exists u_3 \in \mathcal{O}^\times _K\) and \(\exists Z \in \mathcal{O}_K\) such that \(z = u_3 Z^3\).

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

Then \(a^3 + b^3 \neq u c^3\).

Let \(a, b, c \in \mathbb {N}\).
Let \(a \neq 0\), \(b \neq 0\) and \(c \neq 0\).

Then \(a^3 + b^3 \neq c^3\).

Let \(a, b, c \in \mathbb {N}\).
Let \(3 \nmid abc\).

Then \(a ^3 + b ^3 \neq c ^3\).

To prove Theorem 3.66, it suffices to prove that

Equivalently,

implies Theorem 3.66.

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 \(\mathcal{O}^\times _K\) 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 \(u \in \mathcal{O}^\times _K\) be a unit.

Then \(u \in \left\{ 1, -1, \eta , -\eta , \eta ^2, -\eta ^2\right\} \).

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 \(\mathcal{O}^\times _K\) 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 \(m \in \mathbb {Z}\).

Then \(3 \nmid \eta - m\).

Let \(p \in \mathbb {N}\) be prime.

If \(\zeta _p\) is a primitive \(p\)-th root of unity, then \(\zeta _p - 1\) is prime.