mathlib documentation

number_theory.pell

Pell's equation and Matiyasevic's theorem #

This file solves Pell's equation, i.e. integer solutions to x ^ 2 - d * y ^ 2 = 1 in the special case that d = a ^ 2 - 1. This is then applied to prove Matiyasevic's theorem that the power function is Diophantine, which is the last key ingredient in the solution to Hilbert's tenth problem. For the definition of Diophantine function, see dioph.lean.

Main definition #

pell is a function assigning to a natural number n the n-th solution to Pell's equation constructed recursively from the initial solution (0, 1).

Main statements #

eq_pell shows that every solution to Pell's equation is recursively obtained using pell
matiyasevic shows that a certain system of Diophantine equations has a solution if and only if the first variable is the x-component in a solution to Pell's equation - the key step towards Hilbert's tenth problem in Davis' version of Matiyasevic's theorem.
eq_pow_of_pell shows that the power function is Diophantine.

Implementation notes #

The proof of Matiyasevic's theorem doesn't follow Matiyasevic's original account of using Fibonacci numbers but instead Davis' variant of using solutions to Pell's equation.

References #

Tags #

Pell's equation, Matiyasevic's theorem, Hilbert's tenth problem

TODO #

Provide solutions to Pell's equation for the case of arbitrary d (not just d = a ^ 2 - 1 like in the current version) and furthermore also for x ^ 2 - d * y ^ 2 = -1.
Connect solutions to the continued fraction expansion of √d.

@[simp]

theorem pell.d_pos {a : ℕ} (a1 : 1 < a) :

0 < d a1

@[nolint]

def pell.pell {a : ℕ} (a1 : 1 < a) :

ℕ → ℕ × ℕ

The Pell sequences, i.e. the sequence of integer solutions to x ^ 2 - d * y ^ 2 = 1, where d = a ^ 2 - 1, defined together in mutual recursion.

Equations

pell.pell a1 = λ (n : ℕ), n.rec_on (1, 0) (λ (n : ℕ) (xy : ℕ × ℕ), (xy.fst * a + d a1 * xy.snd, xy.fst + xy.snd * a))

def pell.xn {a : ℕ} (a1 : 1 < a) (n : ℕ) :

The Pell x sequence.

Equations

pell.xn a1 n = (pell.pell a1 n).fst

def pell.yn {a : ℕ} (a1 : 1 < a) (n : ℕ) :

The Pell y sequence.

Equations

pell.yn a1 n = (pell.pell a1 n).snd

@[simp]

theorem pell.pell_val {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.pell a1 n = (pell.xn a1 n, pell.yn a1 n)

@[simp]

theorem pell.xn_zero {a : ℕ} (a1 : 1 < a) :

pell.xn a1 0 = 1

@[simp]

theorem pell.yn_zero {a : ℕ} (a1 : 1 < a) :

pell.yn a1 0 = 0

@[simp]

theorem pell.xn_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xn a1 (n + 1) = pell.xn a1 n * a + d a1 * pell.yn a1 n

@[simp]

theorem pell.yn_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yn a1 (n + 1) = pell.xn a1 n + pell.yn a1 n * a

@[simp]

theorem pell.xn_one {a : ℕ} (a1 : 1 < a) :

pell.xn a1 1 = a

@[simp]

theorem pell.yn_one {a : ℕ} (a1 : 1 < a) :

pell.yn a1 1 = 1

def pell.xz {a : ℕ} (a1 : 1 < a) (n : ℕ) :

The Pell x sequence, considered as an integer sequence.

Equations

pell.xz a1 n = ↑(pell.xn a1 n)

def pell.yz {a : ℕ} (a1 : 1 < a) (n : ℕ) :

The Pell y sequence, considered as an integer sequence.

Equations

pell.yz a1 n = ↑(pell.yn a1 n)

def pell.az {a : ℕ} :

The element a such that d = a ^ 2 - 1, considered as an integer.

Equations

pell.az = ↑a

theorem pell.asq_pos {a : ℕ} (a1 : 1 < a) :

0 < a * a

theorem pell.dz_val {a : ℕ} (a1 : 1 < a) :

↑(d a1) = pell.az * pell.az - 1

@[simp]

theorem pell.xz_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xz a1 (n + 1) = pell.xz a1 n * pell.az + ↑(d a1) * pell.yz a1 n

@[simp]

theorem pell.yz_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yz a1 (n + 1) = pell.xz a1 n + pell.yz a1 n * pell.az

def pell.pell_zd {a : ℕ} (a1 : 1 < a) (n : ℕ) :

ℤ√↑(d a1)

The Pell sequence can also be viewed as an element of ℤ√d

Equations

pell.pell_zd a1 n = {re := ↑(pell.xn a1 n), im := ↑(pell.yn a1 n)}

@[simp]

theorem pell.pell_zd_re {a : ℕ} (a1 : 1 < a) (n : ℕ) :

(pell.pell_zd a1 n).re = ↑(pell.xn a1 n)

@[simp]

theorem pell.pell_zd_im {a : ℕ} (a1 : 1 < a) (n : ℕ) :

(pell.pell_zd a1 n).im = ↑(pell.yn a1 n)

def pell.is_pell {a : ℕ} (a1 : 1 < a) :

ℤ√↑(d a1) → Prop

The property of being a solution to the Pell equation, expressed as a property of elements of ℤ√d.

Equations

pell.is_pell a1 {re := x, im := y} = (x * x - ↑(d a1) * y * y = 1)

theorem pell.is_pell_nat {a : ℕ} (a1 : 1 < a) {x y : ℕ} :

pell.is_pell a1 {re := ↑x, im := ↑y} ↔ x * x - d a1 * y * y = 1

theorem pell.is_pell_norm {a : ℕ} (a1 : 1 < a) {b : ℤ√↑(d a1)} :

pell.is_pell a1 b ↔ b * b.conj = 1

theorem pell.is_pell_mul {a : ℕ} (a1 : 1 < a) {b c : ℤ√↑(d a1)} (hb : pell.is_pell a1 b) (hc : pell.is_pell a1 c) :

pell.is_pell a1 (b * c)

theorem pell.is_pell_conj {a : ℕ} (a1 : 1 < a) {b : ℤ√↑(d a1)} :

pell.is_pell a1 b ↔ pell.is_pell a1 b.conj

@[simp]

theorem pell.pell_zd_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.pell_zd a1 (n + 1) = pell.pell_zd a1 n * {re := ↑a, im := 1}

theorem pell.is_pell_one {a : ℕ} (a1 : 1 < a) :

pell.is_pell a1 {re := ↑a, im := 1}

theorem pell.is_pell_pell_zd {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.is_pell a1 (pell.pell_zd a1 n)

@[simp]

theorem pell.pell_eqz {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xz a1 n * pell.xz a1 n - ↑(d a1) * pell.yz a1 n * pell.yz a1 n = 1

@[simp]

theorem pell.pell_eq {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xn a1 n * pell.xn a1 n - d a1 * pell.yn a1 n * pell.yn a1 n = 1

@[protected, instance]

def pell.dnsq {a : ℕ} (a1 : 1 < a) :

zsqrtd.nonsquare (d a1)

theorem pell.xn_ge_a_pow {a : ℕ} (a1 : 1 < a) (n : ℕ) :

a ^ n ≤ pell.xn a1 n

theorem pell.n_lt_a_pow {a : ℕ} (a1 : 1 < a) (n : ℕ) :

n < a ^ n

theorem pell.n_lt_xn {a : ℕ} (a1 : 1 < a) (n : ℕ) :

n < pell.xn a1 n

theorem pell.x_pos {a : ℕ} (a1 : 1 < a) (n : ℕ) :

0 < pell.xn a1 n

theorem pell.eq_pell_lem {a : ℕ} (a1 : 1 < a) (n : ℕ) (b : ℤ√↑(d a1)) :

1 ≤ b → pell.is_pell a1 b → b ≤ pell.pell_zd a1 n → (∃ (n : ℕ), b = pell.pell_zd a1 n)

theorem pell.eq_pell_zd {a : ℕ} (a1 : 1 < a) (b : ℤ√↑(d a1)) (b1 : 1 ≤ b) (hp : pell.is_pell a1 b) :

∃ (n : ℕ), b = pell.pell_zd a1 n

theorem pell.eq_pell {a : ℕ} (a1 : 1 < a) {x y : ℕ} (hp : x * x - d a1 * y * y = 1) :

∃ (n : ℕ), x = pell.xn a1 n ∧ y = pell.yn a1 n

Every solution to Pell's equation is recursively obtained from the initial solution (1,0) using the recursion pell.

theorem pell.pell_zd_add {a : ℕ} (a1 : 1 < a) (m n : ℕ) :

pell.pell_zd a1 (m + n) = pell.pell_zd a1 m * pell.pell_zd a1 n

theorem pell.xn_add {a : ℕ} (a1 : 1 < a) (m n : ℕ) :

pell.xn a1 (m + n) = pell.xn a1 m * pell.xn a1 n + d a1 * pell.yn a1 m * pell.yn a1 n

theorem pell.yn_add {a : ℕ} (a1 : 1 < a) (m n : ℕ) :

pell.yn a1 (m + n) = pell.xn a1 m * pell.yn a1 n + pell.yn a1 m * pell.xn a1 n

theorem pell.pell_zd_sub {a : ℕ} (a1 : 1 < a) {m n : ℕ} (h : n ≤ m) :

pell.pell_zd a1 (m - n) = pell.pell_zd a1 m * (pell.pell_zd a1 n).conj

theorem pell.xz_sub {a : ℕ} (a1 : 1 < a) {m n : ℕ} (h : n ≤ m) :

pell.xz a1 (m - n) = pell.xz a1 m * pell.xz a1 n - ↑(d a1) * pell.yz a1 m * pell.yz a1 n

theorem pell.yz_sub {a : ℕ} (a1 : 1 < a) {m n : ℕ} (h : n ≤ m) :

pell.yz a1 (m - n) = pell.xz a1 n * pell.yz a1 m - pell.xz a1 m * pell.yz a1 n

theorem pell.xy_coprime {a : ℕ} (a1 : 1 < a) (n : ℕ) :

(pell.xn a1 n).coprime (pell.yn a1 n)

theorem pell.strict_mono_y {a : ℕ} (a1 : 1 < a) :

strict_mono (pell.yn a1)

theorem pell.strict_mono_x {a : ℕ} (a1 : 1 < a) :

strict_mono (pell.xn a1)

theorem pell.yn_ge_n {a : ℕ} (a1 : 1 < a) (n : ℕ) :

n ≤ pell.yn a1 n

theorem pell.y_mul_dvd {a : ℕ} (a1 : 1 < a) (n k : ℕ) :

pell.yn a1 n ∣ pell.yn a1 (n * k)

theorem pell.y_dvd_iff {a : ℕ} (a1 : 1 < a) (m n : ℕ) :

pell.yn a1 m ∣ pell.yn a1 n ↔ m ∣ n

theorem pell.xy_modeq_yn {a : ℕ} (a1 : 1 < a) (n k : ℕ) :

pell.xn a1 (n * k) ≡ pell.xn a1 n ^ k [MOD pell.yn a1 n ^ 2] ∧ pell.yn a1 (n * k) ≡ k * pell.xn a1 n ^ (k - 1) * pell.yn a1 n [MOD pell.yn a1 n ^ 3]

theorem pell.ysq_dvd_yy {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yn a1 n * pell.yn a1 n ∣ pell.yn a1 (n * pell.yn a1 n)

theorem pell.dvd_of_ysq_dvd {a : ℕ} (a1 : 1 < a) {n t : ℕ} (h : pell.yn a1 n * pell.yn a1 n ∣ pell.yn a1 t) :

pell.yn a1 n ∣ t

theorem pell.pell_zd_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.pell_zd a1 (n + 2) + pell.pell_zd a1 n = ↑(2 * a) * pell.pell_zd a1 (n + 1)

theorem pell.xy_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xn a1 (n + 2) + pell.xn a1 n = 2 * a * pell.xn a1 (n + 1) ∧ pell.yn a1 (n + 2) + pell.yn a1 n = 2 * a * pell.yn a1 (n + 1)

theorem pell.xn_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xn a1 (n + 2) + pell.xn a1 n = 2 * a * pell.xn a1 (n + 1)

theorem pell.yn_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yn a1 (n + 2) + pell.yn a1 n = 2 * a * pell.yn a1 (n + 1)

theorem pell.xz_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.xz a1 (n + 2) = ↑(2 * a) * pell.xz a1 (n + 1) - pell.xz a1 n

theorem pell.yz_succ_succ {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yz a1 (n + 2) = ↑(2 * a) * pell.yz a1 (n + 1) - pell.yz a1 n

theorem pell.yn_modeq_a_sub_one {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yn a1 n ≡ n [MOD a - 1]

theorem pell.yn_modeq_two {a : ℕ} (a1 : 1 < a) (n : ℕ) :

pell.yn a1 n ≡ n [MOD 2]

theorem pell.x_sub_y_dvd_pow_lem (y2 y1 y0 yn1 yn0 xn1 xn0 ay a2 : ℤ) :

(a2 * yn1 - yn0) * ay + y2 - (a2 * xn1 - xn0) = y2 - a2 * y1 + y0 + a2 * (yn1 * ay + y1 - xn1) - (yn0 * ay + y0 - xn0)

theorem pell.x_sub_y_dvd_pow {a : ℕ} (a1 : 1 < a) (y n : ℕ) :

2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ pell.yz a1 n * (↑a - ↑y) + ↑(y ^ n) - pell.xz a1 n

theorem pell.xn_modeq_x2n_add_lem {a : ℕ} (a1 : 1 < a) (n j : ℕ) :

pell.xn a1 n ∣ d a1 * pell.yn a1 n * (pell.yn a1 n * pell.xn a1 j) + pell.xn a1 j

theorem pell.xn_modeq_x2n_add {a : ℕ} (a1 : 1 < a) (n j : ℕ) :

pell.xn a1 (2 * n + j) + pell.xn a1 j ≡ 0 [MOD pell.xn a1 n]

theorem pell.xn_modeq_x2n_sub_lem {a : ℕ} (a1 : 1 < a) {n j : ℕ} (h : j ≤ n) :

pell.xn a1 (2 * n - j) + pell.xn a1 j ≡ 0 [MOD pell.xn a1 n]

theorem pell.xn_modeq_x2n_sub {a : ℕ} (a1 : 1 < a) {n j : ℕ} (h : j ≤ 2 * n) :

pell.xn a1 (2 * n - j) + pell.xn a1 j ≡ 0 [MOD pell.xn a1 n]

theorem pell.xn_modeq_x4n_add {a : ℕ} (a1 : 1 < a) (n j : ℕ) :

pell.xn a1 (4 * n + j) ≡ pell.xn a1 j [MOD pell.xn a1 n]

theorem pell.xn_modeq_x4n_sub {a : ℕ} (a1 : 1 < a) {n j : ℕ} (h : j ≤ 2 * n) :

pell.xn a1 (4 * n - j) ≡ pell.xn a1 j [MOD pell.xn a1 n]

theorem pell.eq_of_xn_modeq_lem1 {a : ℕ} (a1 : 1 < a) {i n j : ℕ} :

i < j → j < n → pell.xn a1 i % pell.xn a1 n < pell.xn a1 j % pell.xn a1 n

theorem pell.eq_of_xn_modeq_lem2 {a : ℕ} (a1 : 1 < a) {n : ℕ} (h : 2 * pell.xn a1 n = pell.xn a1 (n + 1)) :

a = 2 ∧ n = 0

theorem pell.eq_of_xn_modeq_lem3 {a : ℕ} (a1 : 1 < a) {i n : ℕ} (npos : 0 < n) {j : ℕ} :

i < j → j ≤ 2 * n → j ≠ n → ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2) → pell.xn a1 i % pell.xn a1 n < pell.xn a1 j % pell.xn a1 n

theorem pell.eq_of_xn_modeq_le {a : ℕ} (a1 : 1 < a) {i j n : ℕ} (ij : i ≤ j) (j2n : j ≤ 2 * n) (h : pell.xn a1 i ≡ pell.xn a1 j [MOD pell.xn a1 n]) (ntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)) :

i = j

theorem pell.eq_of_xn_modeq {a : ℕ} (a1 : 1 < a) {i j n : ℕ} (i2n : i ≤ 2 * n) (j2n : j ≤ 2 * n) (h : pell.xn a1 i ≡ pell.xn a1 j [MOD pell.xn a1 n]) (ntriv : a = 2 → n = 1 → (i = 0 → j ≠ 2) ∧ (i = 2 → j ≠ 0)) :

i = j

theorem pell.eq_of_xn_modeq' {a : ℕ} (a1 : 1 < a) {i j n : ℕ} (ipos : 0 < i) (hin : i ≤ n) (j4n : j ≤ 4 * n) (h : pell.xn a1 j ≡ pell.xn a1 i [MOD pell.xn a1 n]) :

j = i ∨ j + i = 4 * n

theorem pell.modeq_of_xn_modeq {a : ℕ} (a1 : 1 < a) {i j n : ℕ} (ipos : 0 < i) (hin : i ≤ n) (h : pell.xn a1 j ≡ pell.xn a1 i [MOD pell.xn a1 n]) :

j ≡ i [MOD 4 * n] ∨ j + i ≡ 0 [MOD 4 * n]

theorem pell.xy_modeq_of_modeq {a b c : ℕ} (a1 : 1 < a) (b1 : 1 < b) (h : a ≡ b [MOD c]) (n : ℕ) :

pell.xn a1 n ≡ pell.xn b1 n [MOD c] ∧ pell.yn a1 n ≡ pell.yn b1 n [MOD c]

theorem pell.matiyasevic {a k x y : ℕ} :

(∃ (a1 : 1 < a), pell.xn a1 k = x ∧ pell.yn a1 k = y) ↔ 1 < a ∧ k ≤ y ∧ (x = 1 ∧ y = 0 ∨ ∃ (u v s t b : ℕ), x * x - (a * a - 1) * y * y = 1 ∧ u * u - (a * a - 1) * v * v = 1 ∧ s * s - (b * b - 1) * t * t = 1 ∧ 1 < b ∧ b ≡ 1 [MOD 4 * y] ∧ b ≡ a [MOD u] ∧ 0 < v ∧ y * y ∣ v ∧ s ≡ x [MOD u] ∧ t ≡ k [MOD 4 * y])

theorem pell.eq_pow_of_pell_lem {a y k : ℕ} (a1 : 1 < a) (ypos : 0 < y) :

0 < k → y ^ k < a → ↑(y ^ k) < 2 * ↑a * ↑y - ↑y * ↑y - 1

theorem pell.eq_pow_of_pell {m n k : ℕ} :

n ^ k = m ↔ k = 0 ∧ m = 1 ∨ 0 < k ∧ (n = 0 ∧ m = 0 ∨ 0 < n ∧ ∃ (w a t z : ℕ) (a1 : 1 < a), pell.xn a1 k ≡ pell.yn a1 k * (a - n) + m [MOD t] ∧ 2 * a * n = t + (n * n + 1) ∧ m < t ∧ n ≤ w ∧ k ≤ w ∧ a * a - ((w + 1) * (w + 1) - 1) * (w * z) * (w * z) = 1)