FormalConjectures.Wikipedia.DiophantineTuple

FormalConjectures
Wikipedia
DiophantineTuple

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import FormalConjecturesUtil

Diophantine $m$-tuples

A Diophantine $m$-tuple is a set of $m$ distinct positive integers ${a_1, \dots, a_m}$ such that $a_i a_j + 1$ is a perfect square for every $i \neq j$.

References:

Wikipedia

[Du16] Dujella, Andrej, What is... a Diophantine m-tuple?, Notices Amer. Math. Soc. 63 (2016), 772--774.

[Du] Dujella, Andrej, Diophantine quintuple conjecture.

[HTZ19] He, Bo, Togbé, Alain and Ziegler, Volker, There is no Diophantine quintuple, Trans. Amer. Math. Soc. 371 (2019), 6665--6709.

[BD69] Baker, Alan and Davenport, Harold, The equations $3x^2-2=y^2$ and $8x^2-7=z^2$. Quart. J. Math. Oxford Ser. (2) 20 (1969), 129--137.

[Gi99] Gibbs, Philip, Some rational Diophantine sextuples, (1999).

namespace DiophantineTuplevariable {R : Type*} [Semiring R]

A finite set s is a Diophantine tuple if each element is nonzero and the product of any two distinct elements is one less than a perfect square.

We define this for all semirings and specialize to the integral and rational cases in this file; these are the most common in the literature. Note that the cases of ℕ and ℕ+ are mathematically equivalent.

def IsDiophantineTuple (s : Finset R) : Prop :=
  (∀ x ∈ s, x ≠ 0) ∧
  ∀ x ∈ s, ∀ y ∈ s, x ≠ y → IsSquare (x * y + 1)

An integral Diophantine tuple is a Diophantine tuple of positive integers.

abbrev IsIntegralDiophantineTuple (s : Finset ℕ) : Prop := IsDiophantineTuple (R := ℕ) s

A rational Diophantine tuple is a Diophantine tuple of nonzero rationals.

abbrev IsRationalDiophantineTuple (s : Finset ℚ) : Prop := IsDiophantineTuple (R := ℚ) s

The property of being a Diophantine tuple is closed under subsets. [Du16]

@[category textbook, AMS 11]
theorem isDiophantineTuple_of_subset (s t : Finset R) (h1 : IsDiophantineTuple t)
    (h2 : s ⊆ t) : IsDiophantineTuple s := byR:Type u_1inst✝:Semiring Rs:Finset Rt:Finset Rh1:IsDiophantineTuple th2:s ⊆ t⊢ IsDiophantineTuple s 
  sorryAll goals completed! 🐙

There exists a Diophantine 4-tuple; this example is due to Fermat.

@[category test, AMS 11]
theorem fermat_4_tuple : IsIntegralDiophantineTuple {1, 3, 8, 120} := by⊢ IsIntegralDiophantineTuple {1, 3, 8, 120}
  norm_num [IsDiophantineTuple]All goals completed! 🐙

The set of integral Diophantine 4-tuples.

def integralDiophantine4Tuple : Set (Finset ℕ) :=
  { s | IsIntegralDiophantineTuple s ∧ s.card = 4 }

/-- There exist infinitely many integral Diophantine 4-tuples. [Du16]

Proof: many parameterizations exist, e.g. ${k, k + 2, 4k + 4, 16k^3 + 48k^2 + 44k + 12}$.

@[category textbook, AMS 11]
example : integralDiophantine4Tuple.encard = ⊤ := byR:Type u_1inst✝:Semiring R⊢ integralDiophantine4Tuple.encard = ⊤ sorryAll goals completed! 🐙

The statement that there is no integral Diophantine 5-tuple.

abbrev NoIntegralDiophantineFiveTuple := ¬∃ t, IsIntegralDiophantineTuple t ∧ t.card = 5

The Diophantine 5-tuple theorem: there does not exist an integral Diophantine 5-tuple. [HTZ19]

@[category research solved, AMS 11]
theorem noIntegralDiophantineFiveTuple : NoIntegralDiophantineFiveTuple := by⊢ NoIntegralDiophantineFiveTuple sorryAll goals completed! 🐙

Given an integral Diophantine 3-tuple, there is a standard way to extend it to a 4-tuple by adjoining the value of this function. This is $d_+$ in [Du].

Note that when ${a, b, c}$ is a Diophantine tuple, each factor under the square root (e.g. $ab + 1$) is a perfect square.

def regularExtension (a b c : ℕ) : ℕ :=
  a + b + c + 2 * a * b * c + 2 * Nat.sqrt ((a * b + 1) * (a * c + 1) * (b * c + 1))

@[category test, AMS 11]
example : regularExtension 1 3 8 = 120 := byR:Type u_1inst✝:Semiring R⊢ regularExtension 1 3 8 = 120 norm_num [regularExtension]All goals completed! 🐙

The property that the Diophantine 3-tuple ${a, b, c}$ extends to a 4-tuple ${a, b, c, d}$ with $d > \max(a, b, c)$ only via regularExtension.

def HasUniqueExtension (a b c : ℕ) : Prop :=
  (IsIntegralDiophantineTuple { a, b, c }) ∧
  ∀ d : ℕ, (IsIntegralDiophantineTuple { a, b, c, d }) → max a (max b c) < d →
    d = regularExtension a b c

The statement that every integral Diophantine triple of three distinct elements has a unique extension by a larger element.

abbrev HasUniqueExtensionOfForall :=
  ∀ a b c : ℕ, ({a, b, c} : Finset ℕ).card = 3 → IsIntegralDiophantineTuple {a, b, c} →
    HasUniqueExtension a b c

The "strong Diophantine 5-tuple conjecture", so-called because it implies the Diophantine 5-tuple theorem (see noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall). [Du]

@[category research open, AMS 11]
theorem hasUniqueExtension_of_forall : HasUniqueExtensionOfForall := by⊢ HasUniqueExtensionOfForall sorryAll goals completed! 🐙

If every integral Diophantine triple has a unique extension by a larger element, then there is no integral Diophantine 5-tuple. [Du]

Proof: suppose a 5-tuple $a < b < c < d_1 < d_2$ exists; then $d_1 = d_2$ by unique extension.

@[category textbook, AMS 11]
theorem noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall :
    HasUniqueExtensionOfForall → NoIntegralDiophantineFiveTuple := by⊢ HasUniqueExtensionOfForall → NoIntegralDiophantineFiveTuple sorryAll goals completed! 🐙

HasUniqueExtension is known to hold for certain triples, including ${1, 3, 8}$: this is essentially the Baker–Davenport theorem [BD69], which states that $120$ is the only integer $d$ such that ${1, 3, 8, d}$ is a Diophantine tuple. Together with a finite check ruling out $d < 8$, this shows that no integral Diophantine tuple properly extends ${1, 3, 8, 120}$.

@[category research solved, AMS 11]
theorem hasUniqueExtension_of_1_3_8 : HasUniqueExtension 1 3 8 := by⊢ HasUniqueExtension 1 3 8 sorryAll goals completed! 🐙

/-
Theorems and conjectures about the rational case
-/

/-- An example due to Euler which extends fermat_4_tuple, showing that hasUniqueExtension_of_1_3_8 requires integrality.

@[category test, AMS 11]
example : IsRationalDiophantineTuple {1, 3, 8, 120, 777480/8288641} := byR:Type u_1inst✝:Semiring R⊢ IsRationalDiophantineTuple {1, 3, 8, 120, 777480 / 8288641}
  norm_num [IsDiophantineTuple]All goals completed! 🐙

A known rational Diophantine 6-tuple. [Gi99]

@[category test, AMS 11]
theorem gibbs_6_tuple :
    IsRationalDiophantineTuple {11/192, 35/192, 155/27, 512/27, 1235/48, 180873/16} := by⊢ IsRationalDiophantineTuple {11 / 192, 35 / 192, 155 / 27, 512 / 27, 1235 / 48, 180873 / 16}
  norm_num [IsDiophantineTuple]All goals completed! 🐙

Does there exist a rational Diophantine 7-tuple? [Du16]

@[category research open, AMS 11]
theorem rational_7_tuple :
    answer(sorry) ↔ ∃ t, IsRationalDiophantineTuple t ∧ t.card = 7 := by⊢ True ↔ ∃ t, IsRationalDiophantineTuple t ∧ t.card = 7
  sorryAll goals completed! 🐙

end DiophantineTuple