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-/
import FormalConjecturesUtil
namespace Buchi
IsBuchi M asserts that whenever M consecutive values (x + n)² + a (for
n = 0, …, M - 1) are all perfect squares, then a must be 0.
def IsBuchi (M : ℕ) : Prop :=
∀ x a : ℤ, (∀ n ∈ Finset.range M, IsSquare ((x + (n : ℤ)) ^ 2 + a)) → a = 0
Büchi's problem There exists a positive integer $M$ such that, for all integers $x$ and $a$, if $(x+n)^2 + a$ is a square for $M$ consecutive values of $n$, then $a = 0$.
@[category research open, AMS 11]
theorem buchi_problem :
answer(sorry) ↔ ∃ M : ℕ, 1 ≤ M ∧ IsBuchi M := ⊢ True ↔ ∃ M, 1 ≤ M ∧ IsBuchi M
All goals completed! 🐙
Büchi's problem (first open case, $M = 5$): For all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$, then $a = 0$.
Non-trivial sequences of length 3 and 4 are known to exist, so $M = 5$ is the first open case.
@[category research open, AMS 11]
theorem buchi_problem_M5 : IsBuchi 5 := ⊢ IsBuchi 5
All goals completed! 🐙The case $M = 0$ fails: the hypothesis is vacuous, so $a = 0$ cannot be concluded.
@[category test, AMS 11]
theorem buchi_false_M0 : ¬ IsBuchi 0 := ⊢ ¬IsBuchi 0
⊢ ∃ x x_1, (∀ n ∈ ∅, IsSquare ((x + ↑n) ^ 2 + x_1)) ∧ ¬x_1 = 0
exact ⟨0, 1, ⊢ (∀ n ∈ ∅, IsSquare ((0 + ↑n) ^ 2 + 1)) ∧ ¬1 = 0 All goals completed! 🐙⟩The case $M = 1$ fails: $0^2 + 4 = 4 = 2^2$ is a square but $a = 4 \neq 0$.
@[category test, AMS 11]
theorem buchi_false_M1 : ¬ IsBuchi 1 := ⊢ ¬IsBuchi 1
⊢ ∃ x x_1, ∃ (_ : ∀ n ∈ Finset.range 1, IsSquare ((x + ↑n) ^ 2 + x_1)), ¬x_1 = 0
refine ⟨0, 4, ?_, ⊢ ¬4 = 0 All goals completed! 🐙⟩
intro n n:ℕhn:n ∈ Finset.range 1⊢ IsSquare ((0 + ↑n) ^ 2 + 4)
n:ℕhn:n = 0⊢ IsSquare ((0 + ↑n) ^ 2 + 4); ⊢ IsSquare ((0 + ↑0) ^ 2 + 4)
exact ⟨2, ⊢ (0 + ↑0) ^ 2 + 4 = 2 * 2 All goals completed! 🐙⟩The case $M = 2$ fails: $(7+n)^2 - 48$ is a perfect square for $n = 0, 1$ ($1^2$ and $4^2$), but $a = -48 \neq 0$.
@[category test, AMS 11]
theorem buchi_false_M2 : ¬ IsBuchi 2 := ⊢ ¬IsBuchi 2
⊢ ∃ x x_1, ∃ (_ : ∀ n ∈ Finset.range 2, IsSquare ((x + ↑n) ^ 2 + x_1)), ¬x_1 = 0
refine ⟨7, -48, ?_, ⊢ ¬-48 = 0 All goals completed! 🐙⟩
intro n n:ℕhn:n ∈ Finset.range 2⊢ IsSquare ((7 + ↑n) ^ 2 + -48)
n:ℕhn:n < 2⊢ IsSquare ((7 + ↑n) ^ 2 + -48)
n:ℕhn:0 < 2⊢ IsSquare ((7 + ↑0) ^ 2 + -48)n:ℕhn:1 < 2⊢ IsSquare ((7 + ↑1) ^ 2 + -48)
n:ℕhn:0 < 2⊢ IsSquare ((7 + ↑0) ^ 2 + -48) exact ⟨1, n:ℕhn:0 < 2⊢ (7 + ↑0) ^ 2 + -48 = 1 * 1 All goals completed! 🐙⟩
n:ℕhn:1 < 2⊢ IsSquare ((7 + ↑1) ^ 2 + -48) exact ⟨4, n:ℕhn:1 < 2⊢ (7 + ↑1) ^ 2 + -48 = 4 * 4 All goals completed! 🐙⟩The case $M = 3$ fails: $(24+n)^2 - 576$ is a perfect square for $n = 0, 1, 2$ ($0^2$, $7^2$, and $10^2$), but $a = -576 \neq 0$.
@[category test, AMS 11]
theorem buchi_false_M3 : ¬ IsBuchi 3 := ⊢ ¬IsBuchi 3
⊢ ∃ x x_1, ∃ (_ : ∀ n ∈ Finset.range 3, IsSquare ((x + ↑n) ^ 2 + x_1)), ¬x_1 = 0
refine ⟨24, -576, ?_, ⊢ ¬-576 = 0 All goals completed! 🐙⟩
intro n n:ℕhn:n ∈ Finset.range 3⊢ IsSquare ((24 + ↑n) ^ 2 + -576)
n:ℕhn:n < 3⊢ IsSquare ((24 + ↑n) ^ 2 + -576)
n:ℕhn:0 < 3⊢ IsSquare ((24 + ↑0) ^ 2 + -576)n:ℕhn:1 < 3⊢ IsSquare ((24 + ↑1) ^ 2 + -576)n:ℕhn:2 < 3⊢ IsSquare ((24 + ↑2) ^ 2 + -576)
n:ℕhn:0 < 3⊢ IsSquare ((24 + ↑0) ^ 2 + -576) exact ⟨0, n:ℕhn:0 < 3⊢ (24 + ↑0) ^ 2 + -576 = 0 * 0 All goals completed! 🐙⟩
n:ℕhn:1 < 3⊢ IsSquare ((24 + ↑1) ^ 2 + -576) exact ⟨7, n:ℕhn:1 < 3⊢ (24 + ↑1) ^ 2 + -576 = 7 * 7 All goals completed! 🐙⟩
n:ℕhn:2 < 3⊢ IsSquare ((24 + ↑2) ^ 2 + -576) exact ⟨10, n:ℕhn:2 < 3⊢ (24 + ↑2) ^ 2 + -576 = 10 * 10 All goals completed! 🐙⟩The case $M = 4$ fails: $(246+n)^2 - 60480$ is a perfect square for $n = 0, 1, 2, 3$ ($6^2$, $23^2$, $32^2$, and $39^2$), but $a = -60480 \neq 0$. The sequence $(6, 23, 32, 39)$ is a nontrivial Büchi sequence of length 4 (Hensley 1983).
@[category test, AMS 11]
theorem buchi_false_M4 : ¬ IsBuchi 4 := ⊢ ¬IsBuchi 4
⊢ ∃ x x_1, ∃ (_ : ∀ n ∈ Finset.range 4, IsSquare ((x + ↑n) ^ 2 + x_1)), ¬x_1 = 0
refine ⟨246, -60480, ?_, ⊢ ¬-60480 = 0 All goals completed! 🐙⟩
intro n n:ℕhn:n ∈ Finset.range 4⊢ IsSquare ((246 + ↑n) ^ 2 + -60480)
n:ℕhn:n < 4⊢ IsSquare ((246 + ↑n) ^ 2 + -60480)
n:ℕhn:0 < 4⊢ IsSquare ((246 + ↑0) ^ 2 + -60480)n:ℕhn:1 < 4⊢ IsSquare ((246 + ↑1) ^ 2 + -60480)n:ℕhn:2 < 4⊢ IsSquare ((246 + ↑2) ^ 2 + -60480)n:ℕhn:3 < 4⊢ IsSquare ((246 + ↑3) ^ 2 + -60480)
n:ℕhn:0 < 4⊢ IsSquare ((246 + ↑0) ^ 2 + -60480) exact ⟨6, n:ℕhn:0 < 4⊢ (246 + ↑0) ^ 2 + -60480 = 6 * 6 All goals completed! 🐙⟩
n:ℕhn:1 < 4⊢ IsSquare ((246 + ↑1) ^ 2 + -60480) exact ⟨23, n:ℕhn:1 < 4⊢ (246 + ↑1) ^ 2 + -60480 = 23 * 23 All goals completed! 🐙⟩
n:ℕhn:2 < 4⊢ IsSquare ((246 + ↑2) ^ 2 + -60480) exact ⟨32, n:ℕhn:2 < 4⊢ (246 + ↑2) ^ 2 + -60480 = 32 * 32 All goals completed! 🐙⟩
n:ℕhn:3 < 4⊢ IsSquare ((246 + ↑3) ^ 2 + -60480) exact ⟨39, n:ℕhn:3 < 4⊢ (246 + ↑3) ^ 2 + -60480 = 39 * 39 All goals completed! 🐙⟩
end Buchi