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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dchrfi 27101 | The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐷 ∈ Fin) | ||
Theorem | dchrghm 27102 | A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) | ||
Theorem | dchr1 27103 | Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → ( 1 ‘𝐴) = 1) | ||
Theorem | dchreq 27104* | A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 = 𝑌 ↔ ∀𝑘 ∈ 𝑈 (𝑋‘𝑘) = (𝑌‘𝑘))) | ||
Theorem | dchrresb 27105 | A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → ((𝑋 ↾ 𝑈) = (𝑌 ↾ 𝑈) ↔ 𝑋 = 𝑌)) | ||
Theorem | dchrabs 27106 | A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) ⇒ ⊢ (𝜑 → (abs‘(𝑋‘𝐴)) = 1) | ||
Theorem | dchrinv 27107 | The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of 𝑋 are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝜑 → (𝐼‘𝑋) = (∗ ∘ 𝑋)) | ||
Theorem | dchrabs2 27108 | A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → (abs‘(𝑋‘𝐴)) ≤ 1) | ||
Theorem | dchr1re 27109 | The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (0g‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 1 :𝐵⟶ℝ) | ||
Theorem | dchrptlem1 27110* | Lemma for dchrpt 27113. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) & ⊢ 𝑃 = (𝐻dProj𝑆) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊‘𝐼)))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝑊) & ⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 ) & ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)))) ⇒ ⊢ (((𝜑 ∧ 𝐶 ∈ 𝑈) ∧ (𝑀 ∈ ℤ ∧ ((𝑃‘𝐼)‘𝐶) = (𝑀 · (𝑊‘𝐼)))) → (𝑋‘𝐶) = (𝑇↑𝑀)) | ||
Theorem | dchrptlem2 27111* | Lemma for dchrpt 27113. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) & ⊢ 𝑃 = (𝐻dProj𝑆) & ⊢ 𝑂 = (od‘𝐻) & ⊢ 𝑇 = (-1↑𝑐(2 / (𝑂‘(𝑊‘𝐼)))) & ⊢ (𝜑 → 𝐼 ∈ dom 𝑊) & ⊢ (𝜑 → ((𝑃‘𝐼)‘𝐴) ≠ 1 ) & ⊢ 𝑋 = (𝑢 ∈ 𝑈 ↦ (℩ℎ∃𝑚 ∈ ℤ (((𝑃‘𝐼)‘𝑢) = (𝑚 · (𝑊‘𝐼)) ∧ ℎ = (𝑇↑𝑚)))) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrptlem3 27112* | Lemma for dchrpt 27113. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) & ⊢ · = (.g‘𝐻) & ⊢ 𝑆 = (𝑘 ∈ dom 𝑊 ↦ ran (𝑛 ∈ ℤ ↦ (𝑛 · (𝑊‘𝑘)))) & ⊢ (𝜑 → 𝐴 ∈ 𝑈) & ⊢ (𝜑 → 𝑊 ∈ Word 𝑈) & ⊢ (𝜑 → 𝐻dom DProd 𝑆) & ⊢ (𝜑 → (𝐻 DProd 𝑆) = 𝑈) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrpt 27113* | For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 1 = (1r‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ≠ 1 ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐷 (𝑥‘𝐴) ≠ 1) | ||
Theorem | dchrsum2 27114* | An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝑈 = (Unit‘𝑍) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝑈 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | dchrsum 27115* | An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 1 = (0g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐵 = (Base‘𝑍) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | sumdchr2 27116* | Lemma for sumdchr 27118. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (1r‘𝑍) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)) | ||
Theorem | dchrhash 27117 | There are exactly ϕ(𝑁) Dirichlet characters modulo 𝑁. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) | ||
Theorem | sumdchr 27118* | An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 1 = (1r‘𝑍) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0)) | ||
Theorem | dchr2sum 27119* | An orthogonality relation for Dirichlet characters: the sum of 𝑋(𝑎) · ∗𝑌(𝑎) over all 𝑎 is nonzero only when 𝑋 = 𝑌. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 ((𝑋‘𝑎) · (∗‘(𝑌‘𝑎))) = if(𝑋 = 𝑌, (ϕ‘𝑁), 0)) | ||
Theorem | sum2dchr 27120* | An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑈) ⇒ ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 ((𝑥‘𝐴) · (∗‘(𝑥‘𝐶))) = if(𝐴 = 𝐶, (ϕ‘𝑁), 0)) | ||
Theorem | bcctr 27121 | Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) | ||
Theorem | pcbcctr 27122* | Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = Σ𝑘 ∈ (1...(2 · 𝑁))((⌊‘((2 · 𝑁) / (𝑃↑𝑘))) − (2 · (⌊‘(𝑁 / (𝑃↑𝑘)))))) | ||
Theorem | bcmono 27123 | The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐵 ∈ (ℤ≥‘𝐴) ∧ 𝐵 ≤ (𝑁 / 2)) → (𝑁C𝐴) ≤ (𝑁C𝐵)) | ||
Theorem | bcmax 27124 | The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐾 ∈ ℤ) → ((2 · 𝑁)C𝐾) ≤ ((2 · 𝑁)C𝑁)) | ||
Theorem | bcp1ctr 27125 | Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((2 · (𝑁 + 1))C(𝑁 + 1)) = (((2 · 𝑁)C𝑁) · (2 · (((2 · 𝑁) + 1) / (𝑁 + 1))))) | ||
Theorem | bclbnd 27126 | A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ (𝑁 ∈ (ℤ≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁)) | ||
Theorem | efexple 27127 | Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ 𝑁 ∈ ℤ ∧ 𝐵 ∈ ℝ+) → ((𝐴↑𝑁) ≤ 𝐵 ↔ 𝑁 ≤ (⌊‘((log‘𝐵) / (log‘𝐴))))) | ||
Theorem | bpos1lem 27128* | Lemma for bpos1 27129. (Contributed by Mario Carneiro, 12-Mar-2014.) |
⊢ (∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁)) → 𝜑) & ⊢ (𝑁 ∈ (ℤ≥‘𝑃) → 𝜑) & ⊢ 𝑃 ∈ ℙ & ⊢ 𝐴 ∈ ℕ0 & ⊢ (𝐴 · 2) = 𝐵 & ⊢ 𝐴 < 𝑃 & ⊢ (𝑃 < 𝐵 ∨ 𝑃 = 𝐵) ⇒ ⊢ (𝑁 ∈ (ℤ≥‘𝐴) → 𝜑) | ||
Theorem | bpos1 27129* | Bertrand's postulate, checked numerically for 𝑁 ≤ 64, using the prime sequence 2, 3, 5, 7, 13, 23, 43, 83. (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑁 ≤ ;64) → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) | ||
Theorem | bposlem1 27130 | An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ) → (𝑃↑(𝑃 pCnt ((2 · 𝑁)C𝑁))) ≤ (2 · 𝑁)) | ||
Theorem | bposlem2 27131 | There are no odd primes in the range (2𝑁 / 3, 𝑁] dividing the 𝑁-th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.) |
⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝑃 ∈ ℙ) & ⊢ (𝜑 → 2 < 𝑃) & ⊢ (𝜑 → ((2 · 𝑁) / 3) < 𝑃) & ⊢ (𝜑 → 𝑃 ≤ 𝑁) ⇒ ⊢ (𝜑 → (𝑃 pCnt ((2 · 𝑁)C𝑁)) = 0) | ||
Theorem | bposlem3 27132* | Lemma for bpos 27139. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 27131 and prmfac1 16665, respectively, and it does not have any in the range (𝑁, 2𝑁] by hypothesis, the product of the primes up through 2𝑁 / 3 must be sufficient to compose the whole binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) ⇒ ⊢ (𝜑 → (seq1( · , 𝐹)‘𝐾) = ((2 · 𝑁)C𝑁)) | ||
Theorem | bposlem4 27133* | Lemma for bpos 27139. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) & ⊢ 𝑀 = (⌊‘(√‘(2 · 𝑁))) ⇒ ⊢ (𝜑 → 𝑀 ∈ (3...𝐾)) | ||
Theorem | bposlem5 27134* | Lemma for bpos 27139. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) & ⊢ 𝑀 = (⌊‘(√‘(2 · 𝑁))) ⇒ ⊢ (𝜑 → (seq1( · , 𝐹)‘𝑀) ≤ ((2 · 𝑁)↑𝑐(((√‘(2 · 𝑁)) / 3) + 2))) | ||
Theorem | bposlem6 27135* | Lemma for bpos 27139. By using the various bounds at our disposal, arrive at an inequality that is false for 𝑁 large enough. (Contributed by Mario Carneiro, 14-Mar-2014.) (Revised by Wolf Lammen, 12-Sep-2020.) |
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘5)) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (𝑛↑(𝑛 pCnt ((2 · 𝑁)C𝑁))), 1)) & ⊢ 𝐾 = (⌊‘((2 · 𝑁) / 3)) & ⊢ 𝑀 = (⌊‘(√‘(2 · 𝑁))) ⇒ ⊢ (𝜑 → ((4↑𝑁) / 𝑁) < (((2 · 𝑁)↑𝑐(((√‘(2 · 𝑁)) / 3) + 2)) · (2↑𝑐(((4 · 𝑁) / 3) − 5)))) | ||
Theorem | bposlem7 27136* | Lemma for bpos 27139. The function 𝐹 is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛))))) & ⊢ 𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥)) & ⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → (e↑2) ≤ 𝐴) & ⊢ (𝜑 → (e↑2) ≤ 𝐵) ⇒ ⊢ (𝜑 → (𝐴 < 𝐵 → (𝐹‘𝐵) < (𝐹‘𝐴))) | ||
Theorem | bposlem8 27137 | Lemma for bpos 27139. Evaluate 𝐹(64) and show it is less than log2. (Contributed by Mario Carneiro, 14-Mar-2014.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛))))) & ⊢ 𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥)) ⇒ ⊢ ((𝐹‘;64) ∈ ℝ ∧ (𝐹‘;64) < (log‘2)) | ||
Theorem | bposlem9 27138* | Lemma for bpos 27139. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.) (Proof shortened by AV, 15-Sep-2021.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((((√‘2) · (𝐺‘(√‘𝑛))) + ((9 / 4) · (𝐺‘(𝑛 / 2)))) + ((log‘2) / (√‘(2 · 𝑛))))) & ⊢ 𝐺 = (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) / 𝑥)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → ;64 < 𝑁) & ⊢ (𝜑 → ¬ ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | bpos 27139* | Bertrand's postulate: there is a prime between 𝑁 and 2𝑁 for every positive integer 𝑁. This proof follows Erdős's method, for the most part, but with some refinements due to Shigenori Tochiori to save us some calculations of large primes. See http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate for an overview of the proof strategy. This is Metamath 100 proof #98. (Contributed by Mario Carneiro, 14-Mar-2014.) |
⊢ (𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ 𝑝 ≤ (2 · 𝑁))) | ||
If the congruence ((𝑥↑2) mod 𝑝) = (𝑛 mod 𝑝) has a solution we say that 𝑛 is a quadratic residue mod 𝑝. If the congruence has no solution we say that 𝑛 is a quadratic nonresidue mod 𝑝, see definition in [ApostolNT] p. 178. The Legendre symbol (𝑛 /L 𝑝) is defined in a way that its value is 1 if 𝑛 is a quadratic residue mod 𝑝 and -1 if 𝑛 is a quadratic nonresidue mod 𝑝 (and 0 if 𝑝 divides 𝑛), see lgsqr 27197. Originally, the Legendre symbol (𝑁 /L 𝑃) was defined for odd primes 𝑃 only (and arbitrary integers 𝑁) by Adrien-Marie Legendre in 1798, see definition in [ApostolNT] p. 179. It was generalized to be defined for any positive odd integer by Carl Gustav Jacob Jacobi in 1837 (therefore called "Jacobi symbol" since then), see definition in [ApostolNT] p. 188. Finally, it was generalized to be defined for any integer by Leopold Kronecker in 1885 (therefore called "Kronecker symbol" since then). The definition df-lgs 27141 for the "Legendre symbol" /L is actually the definition of the "Kronecker symbol". Since only one definition (and one class symbol) are provided in set.mm, the names "Legendre symbol", "Jacobi symbol" and "Kronecker symbol" are used synonymously for /L, but mostly it is called "Legendre symbol", even if it is used in the context of a "Jacobi symbol" or "Kronecker symbol". | ||
Syntax | clgs 27140 | Extend class notation with the Legendre symbol function. |
class /L | ||
Definition | df-lgs 27141* | Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers, and also the Jacobi symbol, which restricts the Kronecker symbol to positive odd integers). See definition in [ApostolNT] p. 179 resp. definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ /L = (𝑎 ∈ ℤ, 𝑛 ∈ ℤ ↦ if(𝑛 = 0, if((𝑎↑2) = 1, 1, 0), (if((𝑛 < 0 ∧ 𝑎 < 0), -1, 1) · (seq1( · , (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, (if(𝑚 = 2, if(2 ∥ 𝑎, 0, if((𝑎 mod 8) ∈ {1, 7}, 1, -1)), ((((𝑎↑((𝑚 − 1) / 2)) + 1) mod 𝑚) − 1))↑(𝑚 pCnt 𝑛)), 1)))‘(abs‘𝑛))))) | ||
Theorem | zabsle1 27142 | {-1, 0, 1} is the set of all integers with absolute value at most 1. (Contributed by AV, 13-Jul-2021.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ {-1, 0, 1} ↔ (abs‘𝑍) ≤ 1)) | ||
Theorem | lgslem1 27143 | When 𝑎 is coprime to the prime 𝑝, 𝑎↑((𝑝 − 1) / 2) is equivalent mod 𝑝 to 1 or -1, and so adding 1 makes it equivalent to 0 or 2. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ ¬ 𝑃 ∥ 𝐴) → (((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) ∈ {0, 2}) | ||
Theorem | lgslem2 27144 | The set 𝑍 of all integers with absolute value at most 1 contains {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ (-1 ∈ 𝑍 ∧ 0 ∈ 𝑍 ∧ 1 ∈ 𝑍) | ||
Theorem | lgslem3 27145* | The set 𝑍 of all integers with absolute value at most 1 is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ ((𝐴 ∈ 𝑍 ∧ 𝐵 ∈ 𝑍) → (𝐴 · 𝐵) ∈ 𝑍) | ||
Theorem | lgslem4 27146* | Lemma for lgsfcl2 27149. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 19-Mar-2022.) |
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1) ∈ 𝑍) | ||
Theorem | lgsval 27147* | Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) = if(𝑁 = 0, if((𝐴↑2) = 1, 1, 0), (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁))))) | ||
Theorem | lgsfval 27148* | Value of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ (𝑀 ∈ ℕ → (𝐹‘𝑀) = if(𝑀 ∈ ℙ, (if(𝑀 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑀 − 1) / 2)) + 1) mod 𝑀) − 1))↑(𝑀 pCnt 𝑁)), 1)) | ||
Theorem | lgsfcl2 27149* | The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 27142). (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) & ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶𝑍) | ||
Theorem | lgscllem 27150* | The Legendre symbol is an element of 𝑍. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) & ⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍) | ||
Theorem | lgsfcl 27151* | Closure of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ) | ||
Theorem | lgsfle1 27152* | The function 𝐹 has magnitude less or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ (((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) ∧ 𝑀 ∈ ℕ) → (abs‘(𝐹‘𝑀)) ≤ 1) | ||
Theorem | lgsval2lem 27153* | Lemma for lgsval2 27159. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℙ) → (𝐴 /L 𝑁) = if(𝑁 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑁 − 1) / 2)) + 1) mod 𝑁) − 1))) | ||
Theorem | lgsval4lem 27154* | Lemma for lgsval4 27163. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, (if(𝑛 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑛 − 1) / 2)) + 1) mod 𝑛) − 1))↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1))) | ||
Theorem | lgscl2 27155* | The Legendre symbol is an integer with absolute value less than or equal to 1. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝑍 = {𝑥 ∈ ℤ ∣ (abs‘𝑥) ≤ 1} ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ 𝑍) | ||
Theorem | lgs0 27156 | The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (𝐴 ∈ ℤ → (𝐴 /L 0) = if((𝐴↑2) = 1, 1, 0)) | ||
Theorem | lgscl 27157 | The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | ||
Theorem | lgsle1 27158 | The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 27157 this implies that it takes values in {-1, 0, 1}. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (abs‘(𝐴 /L 𝑁)) ≤ 1) | ||
Theorem | lgsval2 27159 | The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime 2). (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ ℙ) → (𝐴 /L 𝑃) = if(𝑃 = 2, if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1)), ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1))) | ||
Theorem | lgs2 27160 | The Legendre symbol at 2. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (𝐴 ∈ ℤ → (𝐴 /L 2) = if(2 ∥ 𝐴, 0, if((𝐴 mod 8) ∈ {1, 7}, 1, -1))) | ||
Theorem | lgsval3 27161 | The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → (𝐴 /L 𝑃) = ((((𝐴↑((𝑃 − 1) / 2)) + 1) mod 𝑃) − 1)) | ||
Theorem | lgsvalmod 27162 | The Legendre symbol is equivalent to 𝑎↑((𝑝 − 1) / 2), mod 𝑝. This theorem is also called "Euler's criterion", see theorem 9.2 in [ApostolNT] p. 180, or a representation of Euler's criterion using the Legendre symbol, see also lgsqr 27197. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) | ||
Theorem | lgsval4 27163* | Restate lgsval 27147 for nonzero 𝑁, where the function 𝐹 has been abbreviated into a self-referential expression taking the value of /L on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L 𝑁) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · (seq1( · , 𝐹)‘(abs‘𝑁)))) | ||
Theorem | lgsfcl3 27164* | Closure of the function 𝐹 which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → 𝐹:ℕ⟶ℤ) | ||
Theorem | lgsval4a 27165* | Same as lgsval4 27163 for positive 𝑁. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑁)), 1)) ⇒ ⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐴 /L 𝑁) = (seq1( · , 𝐹)‘𝑁)) | ||
Theorem | lgscl1 27166 | The value of the Legendre symbol is either -1 or 0 or 1. (Contributed by AV, 13-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ {-1, 0, 1}) | ||
Theorem | lgsneg 27167 | The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → (𝐴 /L -𝑁) = (if(𝐴 < 0, -1, 1) · (𝐴 /L 𝑁))) | ||
Theorem | lgsneg1 27168 | The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑁 ∈ ℤ) → (𝐴 /L -𝑁) = (𝐴 /L 𝑁)) | ||
Theorem | lgsmod 27169 | The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) → ((𝐴 mod 𝑁) /L 𝑁) = (𝐴 /L 𝑁)) | ||
Theorem | lgsdilem 27170 | Lemma for lgsdi 27180 and lgsdir 27178: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → if((𝑁 < 0 ∧ (𝐴 · 𝐵) < 0), -1, 1) = (if((𝑁 < 0 ∧ 𝐴 < 0), -1, 1) · if((𝑁 < 0 ∧ 𝐵 < 0), -1, 1))) | ||
Theorem | lgsdir2lem1 27171 | Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((1 mod 8) = 1 ∧ (-1 mod 8) = 7) ∧ ((3 mod 8) = 3 ∧ (-3 mod 8) = 5)) | ||
Theorem | lgsdir2lem2 27172 | Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (𝐾 ∈ ℤ ∧ 2 ∥ (𝐾 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝐾) → (𝐴 mod 8) ∈ 𝑆))) & ⊢ 𝑀 = (𝐾 + 1) & ⊢ 𝑁 = (𝑀 + 1) & ⊢ 𝑁 ∈ 𝑆 ⇒ ⊢ (𝑁 ∈ ℤ ∧ 2 ∥ (𝑁 + 1) ∧ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → ((𝐴 mod 8) ∈ (0...𝑁) → (𝐴 mod 8) ∈ 𝑆))) | ||
Theorem | lgsdir2lem3 27173 | Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ ¬ 2 ∥ 𝐴) → (𝐴 mod 8) ∈ ({1, 7} ∪ {3, 5})) | ||
Theorem | lgsdir2lem4 27174 | Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐴 mod 8) ∈ {1, 7}) → (((𝐴 · 𝐵) mod 8) ∈ {1, 7} ↔ (𝐵 mod 8) ∈ {1, 7})) | ||
Theorem | lgsdir2lem5 27175 | Lemma for lgsdir2 27176. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ((𝐴 mod 8) ∈ {3, 5} ∧ (𝐵 mod 8) ∈ {3, 5})) → ((𝐴 · 𝐵) mod 8) ∈ {1, 7}) | ||
Theorem | lgsdir2 27176 | The Legendre symbol is completely multiplicative at 2. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 · 𝐵) /L 2) = ((𝐴 /L 2) · (𝐵 /L 2))) | ||
Theorem | lgsdirprm 27177 | The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 · 𝐵) /L 𝑃) = ((𝐴 /L 𝑃) · (𝐵 /L 𝑃))) | ||
Theorem | lgsdir 27178 | The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that 𝐴 and 𝐵 are odd positive integers). Together with lgsqr 27197 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐴 ≠ 0 ∧ 𝐵 ≠ 0)) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁))) | ||
Theorem | lgsdilem2 27179* | Lemma for lgsdi 27180. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ (𝜑 → 𝑀 ≠ 0) & ⊢ (𝜑 → 𝑁 ≠ 0) & ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ if(𝑛 ∈ ℙ, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑀)), 1)) ⇒ ⊢ (𝜑 → (seq1( · , 𝐹)‘(abs‘𝑀)) = (seq1( · , 𝐹)‘(abs‘(𝑀 · 𝑁)))) | ||
Theorem | lgsdi 27180 | The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that 𝑀 and 𝑁 are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 ≠ 0 ∧ 𝑁 ≠ 0)) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (𝐴 /L 𝑁))) | ||
Theorem | lgsne0 27181 | The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐴 /L 𝑁) ≠ 0 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | lgsabs1 27182 | The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1)) | ||
Theorem | lgssq 27183 | The Legendre symbol at a square is equal to 1. Together with lgsmod 27169 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ 𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1) → ((𝐴↑2) /L 𝑁) = 1) | ||
Theorem | lgssq2 27184 | The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℕ ∧ (𝐴 gcd 𝑁) = 1) → (𝐴 /L (𝑁↑2)) = 1) | ||
Theorem | lgsprme0 27185 | The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ ℙ) → ((𝐴 /L 𝑃) = 0 ↔ (𝐴 mod 𝑃) = 0)) | ||
Theorem | 1lgs 27186 | The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ (𝑁 ∈ ℤ → (1 /L 𝑁) = 1) | ||
Theorem | lgs1 27187 | The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ (𝐴 ∈ ℤ → (𝐴 /L 1) = 1) | ||
Theorem | lgsmodeq 27188 | The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ (𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁)) → ((𝐴 mod 𝑁) = (𝐵 mod 𝑁) → (𝐴 /L 𝑁) = (𝐵 /L 𝑁))) | ||
Theorem | lgsmulsqcoprm 27189 | The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.) |
⊢ (((𝐴 ∈ ℤ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℤ ∧ 𝐵 ≠ 0) ∧ (𝑁 ∈ ℤ ∧ (𝐴 gcd 𝑁) = 1)) → (((𝐴↑2) · 𝐵) /L 𝑁) = (𝐵 /L 𝑁)) | ||
Theorem | lgsdirnn0 27190 | Variation on lgsdir 27178 valid for all 𝐴, 𝐵 but only for positive 𝑁. (The exact location of the failure of this law is for 𝐴 = 0, 𝐵 < 0, 𝑁 = -1 in which case (0 /L -1) = 1 but (𝐵 /L -1) = -1.) (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐴 · 𝐵) /L 𝑁) = ((𝐴 /L 𝑁) · (𝐵 /L 𝑁))) | ||
Theorem | lgsdinn0 27191 | Variation on lgsdi 27180 valid for all 𝑀, 𝑁 but only for positive 𝐴. (The exact location of the failure of this law is for 𝐴 = -1, 𝑀 = 0, and some 𝑁 in which case (-1 /L 0) = 1 but (-1 /L 𝑁) = -1 when -1 is not a quadratic residue mod 𝑁.) (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ ((𝐴 ∈ ℕ0 ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L (𝑀 · 𝑁)) = ((𝐴 /L 𝑀) · (𝐴 /L 𝑁))) | ||
Theorem | lgsqrlem1 27192 | Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝑆 = (Poly1‘𝑌) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐷 = ( deg1 ‘𝑌) & ⊢ 𝑂 = (eval1‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ 𝑋 = (var1‘𝑌) & ⊢ − = (-g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃) = (1 mod 𝑃)) ⇒ ⊢ (𝜑 → ((𝑂‘𝑇)‘(𝐿‘𝐴)) = (0g‘𝑌)) | ||
Theorem | lgsqrlem2 27193* | Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝑆 = (Poly1‘𝑌) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐷 = ( deg1 ‘𝑌) & ⊢ 𝑂 = (eval1‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ 𝑋 = (var1‘𝑌) & ⊢ − = (-g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2))) ⇒ ⊢ (𝜑 → 𝐺:(1...((𝑃 − 1) / 2))–1-1→(◡(𝑂‘𝑇) “ {(0g‘𝑌)})) | ||
Theorem | lgsqrlem3 27194* | Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝑆 = (Poly1‘𝑌) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐷 = ( deg1 ‘𝑌) & ⊢ 𝑂 = (eval1‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ 𝑋 = (var1‘𝑌) & ⊢ − = (-g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → (𝐴 /L 𝑃) = 1) ⇒ ⊢ (𝜑 → (𝐿‘𝐴) ∈ (◡(𝑂‘𝑇) “ {(0g‘𝑌)})) | ||
Theorem | lgsqrlem4 27195* | Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ 𝑌 = (ℤ/nℤ‘𝑃) & ⊢ 𝑆 = (Poly1‘𝑌) & ⊢ 𝐵 = (Base‘𝑆) & ⊢ 𝐷 = ( deg1 ‘𝑌) & ⊢ 𝑂 = (eval1‘𝑌) & ⊢ ↑ = (.g‘(mulGrp‘𝑆)) & ⊢ 𝑋 = (var1‘𝑌) & ⊢ − = (-g‘𝑆) & ⊢ 1 = (1r‘𝑆) & ⊢ 𝑇 = ((((𝑃 − 1) / 2) ↑ 𝑋) − 1 ) & ⊢ 𝐿 = (ℤRHom‘𝑌) & ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2})) & ⊢ 𝐺 = (𝑦 ∈ (1...((𝑃 − 1) / 2)) ↦ (𝐿‘(𝑦↑2))) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → (𝐴 /L 𝑃) = 1) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)) | ||
Theorem | lgsqrlem5 27196* | Lemma for lgsqr 27197. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ (𝐴 /L 𝑃) = 1) → ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)) | ||
Theorem | lgsqr 27197* | The Legendre symbol for odd primes is 1 iff the number is not a multiple of the prime (in which case it is 0, see lgsne0 27181) and the number is a quadratic residue mod 𝑃 (it is -1 for nonresidues by the process of elimination from lgsabs1 27182). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 ↔ (¬ 𝑃 ∥ 𝐴 ∧ ∃𝑥 ∈ ℤ 𝑃 ∥ ((𝑥↑2) − 𝐴)))) | ||
Theorem | lgsqrmod 27198* | If the Legendre symbol of an integer for an odd prime is 1, then the number is a quadratic residue mod 𝑃. (Contributed by AV, 20-Aug-2021.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ ((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃))) | ||
Theorem | lgsqrmodndvds 27199* | If the Legendre symbol of an integer 𝐴 for an odd prime is 1, then the number is a quadratic residue mod 𝑃 with a solution 𝑥 of the congruence (𝑥↑2)≡𝐴 (mod 𝑃) which is not divisible by the prime. (Contributed by AV, 20-Aug-2021.) (Proof shortened by AV, 18-Mar-2022.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) = 1 → ∃𝑥 ∈ ℤ (((𝑥↑2) mod 𝑃) = (𝐴 mod 𝑃) ∧ ¬ 𝑃 ∥ 𝑥))) | ||
Theorem | lgsdchrval 27200* | The Legendre symbol function 𝑋(𝑚) = (𝑚 /L 𝑁), where 𝑁 is an odd positive number, is a Dirichlet character modulo 𝑁. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ 𝑋 = (𝑦 ∈ 𝐵 ↦ (℩ℎ∃𝑚 ∈ ℤ (𝑦 = (𝐿‘𝑚) ∧ ℎ = (𝑚 /L 𝑁)))) ⇒ ⊢ (((𝑁 ∈ ℕ ∧ ¬ 2 ∥ 𝑁) ∧ 𝐴 ∈ ℤ) → (𝑋‘(𝐿‘𝐴)) = (𝐴 /L 𝑁)) |
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