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Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)
TypeLabelDescription
Statement

20.24.21  Pell equations 1: A nontrivial solution always exists

Theorempellexlem1 36301 Lemma for pellex 36307. Arithmetical core of pellexlem3, norm lower bound. This begins Dirichlet's proof of the Pell equation solution existence; the proof here follows theorem 62 of [vandenDries] p. 43. (Contributed by Stefan O'Rear, 14-Sep-2014.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ ¬ (√‘𝐷) ∈ ℚ) → ((𝐴↑2) − (𝐷 · (𝐵↑2))) ≠ 0)

Theorempellexlem2 36302 Lemma for pellex 36307. Arithmetical core of pellexlem3, norm upper bound. (Contributed by Stefan O'Rear, 14-Sep-2014.)
(((𝐷 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) ∧ (abs‘((𝐴 / 𝐵) − (√‘𝐷))) < (𝐵↑-2)) → (abs‘((𝐴↑2) − (𝐷 · (𝐵↑2)))) < (1 + (2 · (√‘𝐷))))

Theorempellexlem3 36303* Lemma for pellex 36307. To each good rational approximation of (√‘𝐷), there exists a near-solution. (Contributed by Stefan O'Rear, 14-Sep-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {𝑥 ∈ ℚ ∣ (0 < 𝑥 ∧ (abs‘(𝑥 − (√‘𝐷))) < ((denom‘𝑥)↑-2))} ≼ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))})

Theorempellexlem4 36304* Lemma for pellex 36307. Invoking irrapx1 36300, we have infinitely many near-solutions. (Contributed by Stefan O'Rear, 14-Sep-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (((𝑦↑2) − (𝐷 · (𝑧↑2))) ≠ 0 ∧ (abs‘((𝑦↑2) − (𝐷 · (𝑧↑2)))) < (1 + (2 · (√‘𝐷)))))} ≈ ℕ)

Theorempellexlem5 36305* Lemma for pellex 36307. Invoking fiphp3d 36291, we have infinitely many near-solutions for some specific norm. (Contributed by Stefan O'Rear, 19-Oct-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℤ (𝑥 ≠ 0 ∧ {⟨𝑦, 𝑧⟩ ∣ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ ((𝑦↑2) − (𝐷 · (𝑧↑2))) = 𝑥)} ≈ ℕ))

Theorempellexlem6 36306* Lemma for pellex 36307. Doing a field division between near solutions get us to norm 1, and the modularity constraint ensures we still have an integer. Returning NN guarantees that we are not returning the trivial solution (1,0). We are not explicitly defining the Pell-field, Pell-ring, and Pell-norm explicitly because after this construction is done we will never use them. This is mostly basic algebraic number theory and could be simplified if a generic framework for that were in place. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(𝜑𝐴 ∈ ℕ)    &   (𝜑𝐵 ∈ ℕ)    &   (𝜑𝐶 ∈ ℤ)    &   (𝜑𝐷 ∈ ℕ)    &   (𝜑 → ¬ (√‘𝐷) ∈ ℚ)    &   (𝜑𝐸 ∈ ℕ)    &   (𝜑𝐹 ∈ ℕ)    &   (𝜑 → ¬ (𝐴 = 𝐸𝐵 = 𝐹))    &   (𝜑𝐶 ≠ 0)    &   (𝜑 → ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 𝐶)    &   (𝜑 → ((𝐸↑2) − (𝐷 · (𝐹↑2))) = 𝐶)    &   (𝜑 → (𝐴 mod (abs‘𝐶)) = (𝐸 mod (abs‘𝐶)))    &   (𝜑 → (𝐵 mod (abs‘𝐶)) = (𝐹 mod (abs‘𝐶)))       (𝜑 → ∃𝑎 ∈ ℕ ∃𝑏 ∈ ℕ ((𝑎↑2) − (𝐷 · (𝑏↑2))) = 1)

Theorempellex 36307* Every Pell equation has a nontrivial solution. Theorem 62 in [vandenDries] p. 43. (Contributed by Stefan O'Rear, 19-Oct-2014.)
((𝐷 ∈ ℕ ∧ ¬ (√‘𝐷) ∈ ℚ) → ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ ((𝑥↑2) − (𝐷 · (𝑦↑2))) = 1)

20.24.22  Pell equations 2: Algebraic number theory of the solution set

Syntaxcsquarenn 36308 Extend class notation to include the set of square positive integers.
class NN

Syntaxcpell1qr 36309 Extend class notation to include the class of quadrant-1 Pell solutions.
class Pell1QR

Syntaxcpell1234qr 36310 Extend class notation to include the class of any-quadrant Pell solutions.
class Pell1234QR

Syntaxcpell14qr 36311 Extend class notation to include the class of positive Pell solutions.
class Pell14QR

Syntaxcpellfund 36312 Extend class notation to include the Pell-equation fundamental solution function.
class PellFund

Definitiondf-squarenn 36313 Define the set of square positive integers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
NN = {𝑥 ∈ ℕ ∣ (√‘𝑥) ∈ ℚ}

Definitiondf-pell1qr 36314* Define the solutions of a Pell equation in the first quadrant. To avoid pair pain, we represent this via the canonical embedding into the reals. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell1QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pell14qr 36315* Define the positive solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell14QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pell1234qr 36316* Define the general solutions of a Pell equation. (Contributed by Stefan O'Rear, 17-Sep-2014.)
Pell1234QR = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝑥) · 𝑤)) ∧ ((𝑧↑2) − (𝑥 · (𝑤↑2))) = 1)})

Definitiondf-pellfund 36317* A function mapping Pell discriminants to the corresponding fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
PellFund = (𝑥 ∈ (ℕ ∖ ◻NN) ↦ inf({𝑧 ∈ (Pell14QR‘𝑥) ∣ 1 < 𝑧}, ℝ, < ))

Theorempell1qrval 36318* Value of the set of first-quadrant Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Theoremelpell1qr 36319* Membership in a first-quadrant Pell solution set. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℕ0 (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Theorempell14qrval 36320* Value of the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Theoremelpell14qr 36321* Membership in the set of positive Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℕ0𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Theorempell1234qrval 36322* Value of the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1234QR‘𝐷) = {𝑦 ∈ ℝ ∣ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝑦 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1)})

Theoremelpell1234qr 36323* Membership in the set of general Pell solutions. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1234QR‘𝐷) ↔ (𝐴 ∈ ℝ ∧ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ (𝐴 = (𝑧 + ((√‘𝐷) · 𝑤)) ∧ ((𝑧↑2) − (𝐷 · (𝑤↑2))) = 1))))

Theorempell1234qrre 36324 General Pell solutions are (coded as) real numbers. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ∈ ℝ)

Theorempell1234qrne0 36325 No solution to a Pell equation is zero. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → 𝐴 ≠ 0)

Theorempell1234qrreccl 36326 General solutions of the Pell equation are closed under reciprocals. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → (1 / 𝐴) ∈ (Pell1234QR‘𝐷))

Theorempell1234qrmulcl 36327 General solutions of the Pell equation are closed under multiplication. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷) ∧ 𝐵 ∈ (Pell1234QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell1234QR‘𝐷))

Theorempell14qrss1234 36328 A positive Pell solution is a general Pell solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell14QR‘𝐷) ⊆ (Pell1234QR‘𝐷))

Theorempell14qrre 36329 A positive Pell solution is a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ)

Theorempell14qrne0 36330 A positive Pell solution is a nonzero number. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ≠ 0)

Theorempell14qrgt0 36331 A positive Pell solution is a positive number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 0 < 𝐴)

Theorempell14qrrp 36332 A positive Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → 𝐴 ∈ ℝ+)

Theorempell1234qrdich 36333 A general Pell solution is either a positive solution, or its negation is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1234QR‘𝐷)) → (𝐴 ∈ (Pell14QR‘𝐷) ∨ -𝐴 ∈ (Pell14QR‘𝐷)))

Theoremelpell14qr2 36334 A number is a positive Pell solution iff it is positive and a Pell solution, justifying our name choice. (Contributed by Stefan O'Rear, 19-Oct-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ (𝐴 ∈ (Pell1234QR‘𝐷) ∧ 0 < 𝐴)))

Theorempell14qrmulcl 36335 Positive Pell solutions are closed under multiplication. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 · 𝐵) ∈ (Pell14QR‘𝐷))

Theorempell14qrreccl 36336 Positive Pell solutions are closed under reciprocal. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (1 / 𝐴) ∈ (Pell14QR‘𝐷))

Theorempell14qrdivcl 36337 Positive Pell solutions are closed under division. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ (Pell14QR‘𝐷)) → (𝐴 / 𝐵) ∈ (Pell14QR‘𝐷))

Theorempell14qrexpclnn0 36338 Lemma for pell14qrexpcl 36339. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℕ0) → (𝐴𝐵) ∈ (Pell14QR‘𝐷))

Theorempell14qrexpcl 36339 Positive Pell solutions are closed under integer powers. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 𝐵 ∈ ℤ) → (𝐴𝐵) ∈ (Pell14QR‘𝐷))

Theorempell1qrss14 36340 First-quadrant Pell solutions are a subset of the positive solutions. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (Pell1QR‘𝐷) ⊆ (Pell14QR‘𝐷))

Theorempell14qrdich 36341 A positive Pell solution is either in the first quadrant, or its reciprocal is. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → (𝐴 ∈ (Pell1QR‘𝐷) ∨ (1 / 𝐴) ∈ (Pell1QR‘𝐷)))

Theorempell1qrge1 36342 A Pell solution in the first quadrant is at least 1. (Contributed by Stefan O'Rear, 17-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷)) → 1 ≤ 𝐴)

Theorempell1qr1 36343 1 is a Pell solution and in the first quadrant as one. (Contributed by Stefan O'Rear, 17-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → 1 ∈ (Pell1QR‘𝐷))

Theoremelpell1qr2 36344 The first quadrant solutions are precisely the positive Pell solutions which are at least one. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell1QR‘𝐷) ↔ (𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 ≤ 𝐴)))

Theorempell1qrgaplem 36345 Lemma for pell1qrgap 36346. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(((𝐷 ∈ ℕ ∧ (𝐴 ∈ ℕ0𝐵 ∈ ℕ0)) ∧ (1 < (𝐴 + ((√‘𝐷) · 𝐵)) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1)) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (𝐴 + ((√‘𝐷) · 𝐵)))

Theorempell1qrgap 36346 First-quadrant Pell solutions are bounded away from 1. (This particular bound allows us to prove exact values for the fundamental solution later.) (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell1QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)

Theorempell14qrgap 36347 Positive Pell solutions are bounded away from 1. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ 𝐴)

Theorempell14qrgapw 36348 Positive Pell solutions are bounded away from 1, with a friendlier bound. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → 2 < 𝐴)

Theorempellqrexplicit 36349 Condition for a calculated real to be a Pell solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℕ0𝐵 ∈ ℕ0) ∧ ((𝐴↑2) − (𝐷 · (𝐵↑2))) = 1) → (𝐴 + ((√‘𝐷) · 𝐵)) ∈ (Pell1QR‘𝐷))

20.24.23  Pell equations 3: characterizing fundamental solution

Theoreminfmrgelbi 36350* Any lower bound of a nonempty set of real numbers is less than or equal to its infimum, one-direction version. (Contributed by Stefan O'Rear, 1-Sep-2013.) (Revised by AV, 17-Sep-2020.)
(((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ) ∧ ∀𝑥𝐴 𝐵𝑥) → 𝐵 ≤ inf(𝐴, ℝ, < ))

Theorempellqrex 36351* There is a nontrivial solution of a Pell equation in the first quadrant. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → ∃𝑥 ∈ (Pell1QR‘𝐷)1 < 𝑥)

Theorempellfundval 36352* Value of the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 18-Sep-2014.) (Revised by AV, 17-Sep-2020.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) = inf({𝑥 ∈ (Pell14QR‘𝐷) ∣ 1 < 𝑥}, ℝ, < ))

Theorempellfundre 36353 The fundamental solution of a Pell equation exists as a real number. (Contributed by Stefan O'Rear, 18-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ)

Theorempellfundge 36354 Lower bound on the fundamental solution of a Pell equation. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → ((√‘(𝐷 + 1)) + (√‘𝐷)) ≤ (PellFund‘𝐷))

Theorempellfundgt1 36355 Weak lower bound on the Pell fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → 1 < (PellFund‘𝐷))

Theorempellfundlb 36356 A nontrivial first quadrant solution is at least as large as the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.) (Proof shortened by AV, 15-Sep-2020.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ 1 < 𝐴) → (PellFund‘𝐷) ≤ 𝐴)

Theorempellfundglb 36357* If a real is larger than the fundamental solution, there is a nontrivial solution less than it. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ ℝ ∧ (PellFund‘𝐷) < 𝐴) → ∃𝑥 ∈ (Pell1QR‘𝐷)((PellFund‘𝐷) ≤ 𝑥𝑥 < 𝐴))

Theorempellfundex 36358 The fundamental solution as an infimum is itself a solution, showing that the solution set is discrete.

Since the fundamental solution is an infimum, there must be an element ge to Fund and lt 2*Fund. If this element is equal to the fundamental solution we're done, otherwise use the infimum again to find another element which must be ge Fund and lt the first element; their ratio is a group element in (1,2), contradicting pell14qrgapw 36348. (Contributed by Stefan O'Rear, 18-Sep-2014.)

(𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ (Pell1QR‘𝐷))

Theorempellfund14gap 36359 There are no solutions between 1 and the fundamental solution. (Contributed by Stefan O'Rear, 18-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷) ∧ (1 ≤ 𝐴𝐴 < (PellFund‘𝐷))) → 𝐴 = 1)

Theorempellfundrp 36360 The fundamental Pell solution is a positive real. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ∈ ℝ+)

Theorempellfundne1 36361 The fundamental Pell solution is never 1. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (PellFund‘𝐷) ≠ 1)

20.24.24  Logarithm laws generalized to an arbitrary base

Section should be obsolete because its contents are covered by section "Logarithms to an arbitrary base" now.

Theoremreglogcl 36362 General logarithm is a real number. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbcl 24198 instead.
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+𝐵 ≠ 1) → ((log‘𝐴) / (log‘𝐵)) ∈ ℝ)

Theoremreglogltb 36363 General logarithm preserves "less than". (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logblt 24209 instead.
(((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴 < 𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) < ((log‘𝐵) / (log‘𝐶))))

Theoremreglogleb 36364 General logarithm preserves . (Contributed by Stefan O'Rear, 19-Oct-2014.) (New usage is discouraged.) Use logbleb 24208 instead.
(((𝐴 ∈ ℝ+𝐵 ∈ ℝ+) ∧ (𝐶 ∈ ℝ+ ∧ 1 < 𝐶)) → (𝐴𝐵 ↔ ((log‘𝐴) / (log‘𝐶)) ≤ ((log‘𝐵) / (log‘𝐶))))

Theoremreglogmul 36365 Multiplication law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbmul 24202 instead.
((𝐴 ∈ ℝ+𝐵 ∈ ℝ+ ∧ (𝐶 ∈ ℝ+𝐶 ≠ 1)) → ((log‘(𝐴 · 𝐵)) / (log‘𝐶)) = (((log‘𝐴) / (log‘𝐶)) + ((log‘𝐵) / (log‘𝐶))))

Theoremreglogexp 36366 Power law for general log. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbzexp 24201 instead.
((𝐴 ∈ ℝ+𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+𝐶 ≠ 1)) → ((log‘(𝐴𝑁)) / (log‘𝐶)) = (𝑁 · ((log‘𝐴) / (log‘𝐶))))

Theoremreglogbas 36367 General log of the base is 1. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logbid1 24193 instead.
((𝐶 ∈ ℝ+𝐶 ≠ 1) → ((log‘𝐶) / (log‘𝐶)) = 1)

Theoremreglog1 36368 General log of 1 is 0. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use logb1 24194 instead.
((𝐶 ∈ ℝ+𝐶 ≠ 1) → ((log‘1) / (log‘𝐶)) = 0)

Theoremreglogexpbas 36369 General log of a power of the base is the exponent. (Contributed by Stefan O'Rear, 19-Sep-2014.) (New usage is discouraged.) Use relogbexp 24205 instead.
((𝑁 ∈ ℤ ∧ (𝐶 ∈ ℝ+𝐶 ≠ 1)) → ((log‘(𝐶𝑁)) / (log‘𝐶)) = 𝑁)

20.24.25  Pell equations 4: the positive solution group is infinite cyclic

Theorempellfund14 36370* Every positive Pell solution is a power of the fundamental solution. (Contributed by Stefan O'Rear, 19-Sep-2014.)
((𝐷 ∈ (ℕ ∖ ◻NN) ∧ 𝐴 ∈ (Pell14QR‘𝐷)) → ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥))

Theorempellfund14b 36371* The positive Pell solutions are precisely the integer powers of the fundamental solution. To get the general solution set (which we will not be using), throw in a copy of Z/2Z. (Contributed by Stefan O'Rear, 19-Sep-2014.)
(𝐷 ∈ (ℕ ∖ ◻NN) → (𝐴 ∈ (Pell14QR‘𝐷) ↔ ∃𝑥 ∈ ℤ 𝐴 = ((PellFund‘𝐷)↑𝑥)))

20.24.26  X and Y sequences 1: Definition and recurrence laws

Syntaxcrmx 36372 Extend class notation to include the Robertson-Matiyasevich X sequence.
class Xrm

Syntaxcrmy 36373 Extend class notation to include the Robertson-Matiyasevich Y sequence.
class Yrm

Definitiondf-rmx 36374* Define the X sequence as the rational part of some solution of a special Pell equation. See frmx 36386 and rmxyval 36388 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Xrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))

Definitiondf-rmy 36375* Define the X sequence as the irrational part of some solution of a special Pell equation. See frmy 36387 and rmxyval 36388 for a more useful but non-eliminable definition. (Contributed by Stefan O'Rear, 21-Sep-2014.)
Yrm = (𝑎 ∈ (ℤ‘2), 𝑛 ∈ ℤ ↦ (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝑎↑2) − 1)) · (2nd𝑏))))‘((𝑎 + (√‘((𝑎↑2) − 1)))↑𝑛))))

Theoremrmxfval 36376* Value of the X sequence. Not used after rmxyval 36388 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm 𝑁) = (1st ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))

Theoremrmyfval 36377* Value of the Y sequence. Not used after rmxyval 36388 is proved. (Contributed by Stefan O'Rear, 21-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Yrm 𝑁) = (2nd ‘((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))))

Theoremrmspecsqrtnq 36378 The discriminant used to define the X and Y sequences has an irrational square root. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof shortened by AV, 2-Aug-2021.)
(𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))

TheoremrmspecsqrtnqOLD 36379 Obsolete version of rmspecsqrtnq 36378 as of 2-Aug-2021. (Contributed by Stefan O'Rear, 21-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴 ∈ (ℤ‘2) → (√‘((𝐴↑2) − 1)) ∈ (ℂ ∖ ℚ))

Theoremrmspecnonsq 36380 The discriminant used to define the X and Y sequences is a nonsquare positive integer and thus a valid Pell equation discriminant. (Contributed by Stefan O'Rear, 21-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → ((𝐴↑2) − 1) ∈ (ℕ ∖ ◻NN))

Theoremqirropth 36381 This lemma implements the concept of "equate rational and irrational parts", used to prove many arithmetical properties of the X and Y sequences. (Contributed by Stefan O'Rear, 21-Sep-2014.)
((𝐴 ∈ (ℂ ∖ ℚ) ∧ (𝐵 ∈ ℚ ∧ 𝐶 ∈ ℚ) ∧ (𝐷 ∈ ℚ ∧ 𝐸 ∈ ℚ)) → ((𝐵 + (𝐴 · 𝐶)) = (𝐷 + (𝐴 · 𝐸)) ↔ (𝐵 = 𝐷𝐶 = 𝐸)))

Theoremrmspecfund 36382 The base of exponent used to define the X and Y sequences is the fundamental solution of the corresponding Pell equation. (Contributed by Stefan O'Rear, 21-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → (PellFund‘((𝐴↑2) − 1)) = (𝐴 + (√‘((𝐴↑2) − 1))))

Theoremrmxyelqirr 36383* The solutions used to construct the X and Y sequences are quadratic irrationals. (Contributed by Stefan O'Rear, 21-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁) ∈ {𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Theoremrmxypairf1o 36384* The function used to extract rational and irrational parts in df-rmx 36374 and df-rmy 36375 in fact achieves a one-to-one mapping from the quadratic irrationals to pairs of integers. (Contributed by Stefan O'Rear, 21-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → (𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏)))):(ℕ0 × ℤ)–1-1-onto→{𝑎 ∣ ∃𝑐 ∈ ℕ0𝑑 ∈ ℤ 𝑎 = (𝑐 + ((√‘((𝐴↑2) − 1)) · 𝑑))})

Theoremrmxyelxp 36385* Lemma for frmx 36386 and frmy 36387. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → ((𝑏 ∈ (ℕ0 × ℤ) ↦ ((1st𝑏) + ((√‘((𝐴↑2) − 1)) · (2nd𝑏))))‘((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁)) ∈ (ℕ0 × ℤ))

Theoremfrmx 36386 The X sequence is a nonnegative integer. See rmxnn 36426 for a strengthening. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Xrm :((ℤ‘2) × ℤ)⟶ℕ0

Theoremfrmy 36387 The Y sequence is an integer. (Contributed by Stefan O'Rear, 22-Sep-2014.)
Yrm :((ℤ‘2) × ℤ)⟶ℤ

Theoremrmxyval 36388 Main definition of the X and Y sequences. Compare definition 2.3 of [JonesMatijasevic] p. 694. (Contributed by Stefan O'Rear, 19-Oct-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm 𝑁) + ((√‘((𝐴↑2) − 1)) · (𝐴 Yrm 𝑁))) = ((𝐴 + (√‘((𝐴↑2) − 1)))↑𝑁))

Theoremrmspecpos 36389 The discriminant used to define the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → ((𝐴↑2) − 1) ∈ ℝ+)

Theoremrmxycomplete 36390* The X and Y sequences taken together enumerate all solutions to the corresponding Pell equation in the right half-plane. This is Metamath 100 proof #39. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑋 ∈ ℕ0𝑌 ∈ ℤ) → (((𝑋↑2) − (((𝐴↑2) − 1) · (𝑌↑2))) = 1 ↔ ∃𝑛 ∈ ℤ (𝑋 = (𝐴 Xrm 𝑛) ∧ 𝑌 = (𝐴 Yrm 𝑛))))

Theoremrmxynorm 36391 The X and Y sequences define a solution to the corresponding Pell equation. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (((𝐴 Xrm 𝑁)↑2) − (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑁)↑2))) = 1)

Theoremrmbaserp 36392 The base of exponentiation for the X and Y sequences is a positive real. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → (𝐴 + (√‘((𝐴↑2) − 1))) ∈ ℝ+)

Theoremrmxyneg 36393 Negation law for X and Y sequences. JonesMatijasevic is inconsistent on whether the X and Y sequences have domain 0 or ; we use consistently to avoid the need for a separate subtraction law. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁) ∧ (𝐴 Yrm -𝑁) = -(𝐴 Yrm 𝑁)))

Theoremrmxyadd 36394 Addition formula for X and Y sequences. See rmxadd 36400 and rmyadd 36404 for most uses. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁)))) ∧ (𝐴 Yrm (𝑀 + 𝑁)) = (((𝐴 Yrm 𝑀) · (𝐴 Xrm 𝑁)) + ((𝐴 Xrm 𝑀) · (𝐴 Yrm 𝑁)))))

Theoremrmxy1 36395 Value of the X and Y sequences at 1. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → ((𝐴 Xrm 1) = 𝐴 ∧ (𝐴 Yrm 1) = 1))

Theoremrmxy0 36396 Value of the X and Y sequences at 0. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → ((𝐴 Xrm 0) = 1 ∧ (𝐴 Yrm 0) = 0))

Theoremrmxneg 36397 Negation law (even function) for the X sequence. The method of proof used for the previous four theorems rmxyneg 36393, rmxyadd 36394, rmxy0 36396, and rmxy1 36395 via qirropth 36381 results in two theorems at once, but typical use requires only one, so this group of theorems serves to separate the cases. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm -𝑁) = (𝐴 Xrm 𝑁))

Theoremrmx0 36398 Value of X sequence at 0. Part 1 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → (𝐴 Xrm 0) = 1)

Theoremrmx1 36399 Value of X sequence at 1. Part 2 of equation 2.11 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
(𝐴 ∈ (ℤ‘2) → (𝐴 Xrm 1) = 𝐴)

Theoremrmxadd 36400 Addition formula for X sequence. Equation 2.7 of [JonesMatijasevic] p. 695. (Contributed by Stefan O'Rear, 22-Sep-2014.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 Xrm (𝑀 + 𝑁)) = (((𝐴 Xrm 𝑀) · (𝐴 Xrm 𝑁)) + (((𝐴↑2) − 1) · ((𝐴 Yrm 𝑀) · (𝐴 Yrm 𝑁)))))

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