Fermat Last Theorem in the case n = 3
#
The goal of this file is to prove Fermat Last theorem in the case n = 3
.
Main results #
fermatLastTheoremThree_case1
: the first case of Fermat Last Theorem whenn = 3
: ifa b c : ℤ
are such that¬ 3 ∣ a * b * c
, thena ^ 3 + b ^ 3 ≠ c ^ 3
.
TODO #
Prove case 2.
To prove Fermat's Last Theorem for n = 3
, it is enough to show that that for all a
, b
, c
in ℤ
such that c ≠ 0
, ¬ 3 ∣ a
, ¬ 3 ∣ b
, a
and b
are coprime and 3 ∣ c
, we have
a ^ 3 + b ^ 3 ≠ c ^ 3
.
FermatLastTheoremForThreeGen
is the statement that a ^ 3 + b ^ 3 = u * c ^ 3
has no
nontrivial solutions in 𝓞 K
for all u : (𝓞 K)ˣ
such that ¬ λ ∣ a
, ¬ λ ∣ b
and λ ∣ c
.
The reason to consider FermatLastTheoremForThreeGen
is to make a descent argument working.
Equations
- One or more equations did not get rendered due to their size.
Instances For
To prove FermatLastTheoremFor 3
, it is enough to prove FermatLastTheoremForThreeGen
.
Solution'
is a tuple given by a solution to a ^ 3 + b ^ 3 = u * c ^ 3
,
where a
, b
, c
and u
are as in FermatLastTheoremForThreeGen
.
See Solution
for the actual structure on which we will do the descent.
- u : (NumberField.RingOfIntegers K)ˣ
- hc : self.c ≠ 0
- coprime : IsCoprime self.a self.b
Instances For
Solution
is the same as Solution'
with the additional assumption that λ ^ 2 ∣ a + b
.
- u : (NumberField.RingOfIntegers K)ˣ
- hc : self.c ≠ 0
- coprime : IsCoprime self.a self.b
Instances For
For any S' : Solution'
, the multiplicity of λ
in S'.c
is finite.
Given S' : Solution'
, S'.multiplicity
is the multiplicity of λ
in S'.c
, as a natural
number.
Equations
- S'.multiplicity = (multiplicity (hζ.toInteger - 1) S'.c).get ⋯
Instances For
Given S : Solution
, S.multiplicity
is the multiplicity of λ
in S.c
, as a natural
number.
Equations
- S.multiplicity = S.multiplicity
Instances For
We say that S : Solution
is minimal if for all S₁ : Solution
, the multiplicity of λ
in
S.c
is less or equal than the multiplicity in S₁.c
.
Equations
- S.isMinimal = ∀ (S₁ : FermatLastTheoremForThreeGen.Solution hζ), S.multiplicity ≤ S₁.multiplicity
Instances For
If there is a solution then there is a minimal one.
Given S' : Solution'
, then S'.a
and S'.b
are both congruent to 1
modulo λ ^ 4
or are
both congruent to -1
.
Given S' : Solution'
, we have that 2 ≤ S'.multiplicity
.
Given S : Solution
, we have that 2 ≤ S.multiplicity
.
Given S' : Solution'
, the key factorization of S'.a ^ 3 + S'.b ^ 3
.
Given S' : Solution'
, we have that λ ^ 2
divides one amongst S'.a + S'.b
,
S'.a + η * S'.b
and S'.a + η ^ 2 * S'.b
.
Given S' : Solution'
, we may assume that λ ^ 2
divides S'.a + S'.b ∨ λ ^ 2
(see also the
result below).
Given S' : Solution'
, then there is S₁ : Solution
such that
S₁.multiplicity = S'.multiplicity
.