![]() |
Metamath
Proof Explorer Theorem List (p. 259 of 454) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-28701) |
![]() (28702-30224) |
![]() (30225-45333) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | logfac2 25801* | Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) = Σ𝑘 ∈ (1...(⌊‘𝐴))((Λ‘𝑘) · (⌊‘(𝐴 / 𝑘)))) | ||
Theorem | chpval2 25802* | Express the second Chebyshev function directly as a sum over the primes less than 𝐴 (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑝 ∈ ((0[,]𝐴) ∩ ℙ)((log‘𝑝) · (⌊‘((log‘𝐴) / (log‘𝑝))))) | ||
Theorem | chpchtsum 25803* | The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.) |
⊢ (𝐴 ∈ ℝ → (ψ‘𝐴) = Σ𝑘 ∈ (1...(⌊‘𝐴))(θ‘(𝐴↑𝑐(1 / 𝑘)))) | ||
Theorem | chpub 25804 | An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.) |
⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴) → (ψ‘𝐴) ≤ ((θ‘𝐴) + ((√‘𝐴) · (log‘𝐴)))) | ||
Theorem | logfacubnd 25805 | A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (log‘(!‘(⌊‘𝐴))) ≤ (𝐴 · (log‘𝐴))) | ||
Theorem | logfaclbnd 25806 | A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.) |
⊢ (𝐴 ∈ ℝ+ → (𝐴 · ((log‘𝐴) − 2)) ≤ (log‘(!‘(⌊‘𝐴)))) | ||
Theorem | logfacbnd3 25807 | Show the stronger statement log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥) alluded to in logfacrlim 25808. (Contributed by Mario Carneiro, 20-May-2016.) |
⊢ ((𝐴 ∈ ℝ+ ∧ 1 ≤ 𝐴) → (abs‘((log‘(!‘(⌊‘𝐴))) − (𝐴 · ((log‘𝐴) − 1)))) ≤ ((log‘𝐴) + 1)) | ||
Theorem | logfacrlim 25808 | Combine the estimates logfacubnd 25805 and logfaclbnd 25806, to get log(𝑥!) = 𝑥log𝑥 + 𝑂(𝑥). Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, log(𝑥!) = 𝑥log𝑥 − 𝑥 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ ((log‘𝑥) − ((log‘(!‘(⌊‘𝑥))) / 𝑥))) ⇝𝑟 1 | ||
Theorem | logexprlim 25809* | The sum Σ𝑛 ≤ 𝑥, log↑𝑁(𝑥 / 𝑛) has the asymptotic expansion (𝑁!)𝑥 + 𝑜(𝑥). (More precisely, the omitted term has order 𝑂(log↑𝑁(𝑥) / 𝑥).) (Contributed by Mario Carneiro, 22-May-2016.) |
⊢ (𝑁 ∈ ℕ0 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛))↑𝑁) / 𝑥)) ⇝𝑟 (!‘𝑁)) | ||
Theorem | logfacrlim2 25810* | Write out logfacrlim 25808 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.) |
⊢ (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((log‘(𝑥 / 𝑛)) / 𝑥)) ⇝𝑟 1 | ||
Theorem | mersenne 25811 | A Mersenne prime is a prime number of the form 2↑𝑃 − 1. This theorem shows that the 𝑃 in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → 𝑃 ∈ ℙ) | ||
Theorem | perfect1 25812 | Euclid's contribution to the Euclid-Euler theorem. A number of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ ((𝑃 ∈ ℤ ∧ ((2↑𝑃) − 1) ∈ ℙ) → (1 σ ((2↑(𝑃 − 1)) · ((2↑𝑃) − 1))) = ((2↑𝑃) · ((2↑𝑃) − 1))) | ||
Theorem | perfectlem1 25813 | Lemma for perfect 25815. (Contributed by Mario Carneiro, 7-Jun-2016.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵))) ⇒ ⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℕ ∧ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈ ℕ)) | ||
Theorem | perfectlem2 25814 | Lemma for perfect 25815. (Contributed by Mario Carneiro, 17-May-2016.) Replace OLD theorem. (Revised by Wolf Lammen, 17-Sep-2020.) |
⊢ (𝜑 → 𝐴 ∈ ℕ) & ⊢ (𝜑 → 𝐵 ∈ ℕ) & ⊢ (𝜑 → ¬ 2 ∥ 𝐵) & ⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵))) ⇒ ⊢ (𝜑 → (𝐵 ∈ ℙ ∧ 𝐵 = ((2↑(𝐴 + 1)) − 1))) | ||
Theorem | perfect 25815* | The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer 𝑁 is a perfect number (that is, its divisor sum is 2𝑁) if and only if it is of the form 2↑(𝑝 − 1) · (2↑𝑝 − 1), where 2↑𝑝 − 1 is prime (a Mersenne prime). (It follows from this that 𝑝 is also prime.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.) |
⊢ ((𝑁 ∈ ℕ ∧ 2 ∥ 𝑁) → ((1 σ 𝑁) = (2 · 𝑁) ↔ ∃𝑝 ∈ ℤ (((2↑𝑝) − 1) ∈ ℙ ∧ 𝑁 = ((2↑(𝑝 − 1)) · ((2↑𝑝) − 1))))) | ||
Syntax | cdchr 25816 | Extend class notation with the group of Dirichlet characters. |
class DChr | ||
Definition | df-dchr 25817* | The group of Dirichlet characters mod 𝑛 is the set of monoid homomorphisms from ℤ / 𝑛ℤ to the multiplicative monoid of the complex numbers, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ DChr = (𝑛 ∈ ℕ ↦ ⦋(ℤ/nℤ‘𝑛) / 𝑧⦌⦋{𝑥 ∈ ((mulGrp‘𝑧) MndHom (mulGrp‘ℂfld)) ∣ (((Base‘𝑧) ∖ (Unit‘𝑧)) × {0}) ⊆ 𝑥} / 𝑏⦌{〈(Base‘ndx), 𝑏〉, 〈(+g‘ndx), ( ∘f · ↾ (𝑏 × 𝑏))〉}) | ||
Theorem | dchrval 25818* | Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) ⇒ ⊢ (𝜑 → 𝐺 = {〈(Base‘ndx), 𝐷〉, 〈(+g‘ndx), ( ∘f · ↾ (𝐷 × 𝐷))〉}) | ||
Theorem | dchrbas 25819* | Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝜑 → 𝐷 = {𝑥 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∣ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑥}) | ||
Theorem | dchrelbas 25820 | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ((𝐵 ∖ 𝑈) × {0}) ⊆ 𝑋))) | ||
Theorem | dchrelbas2 25821* | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈)))) | ||
Theorem | dchrelbas3 25822* | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐷 ↔ (𝑋:𝐵⟶ℂ ∧ (∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 (𝑋‘(𝑥(.r‘𝑍)𝑦)) = ((𝑋‘𝑥) · (𝑋‘𝑦)) ∧ (𝑋‘(1r‘𝑍)) = 1 ∧ ∀𝑥 ∈ 𝐵 ((𝑋‘𝑥) ≠ 0 → 𝑥 ∈ 𝑈))))) | ||
Theorem | dchrelbasd 25823* | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 28-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑁 ∈ ℕ) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ (𝑘 = 𝑥 → 𝑋 = 𝐴) & ⊢ (𝑘 = 𝑦 → 𝑋 = 𝐶) & ⊢ (𝑘 = (𝑥(.r‘𝑍)𝑦) → 𝑋 = 𝐸) & ⊢ (𝑘 = (1r‘𝑍) → 𝑋 = 𝑌) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑈) → 𝑋 ∈ ℂ) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → 𝐸 = (𝐴 · 𝐶)) & ⊢ (𝜑 → 𝑌 = 1) ⇒ ⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 𝑋, 0)) ∈ 𝐷) | ||
Theorem | dchrrcl 25824 | Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) | ||
Theorem | dchrmhm 25825 | A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) ⇒ ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) | ||
Theorem | dchrf 25826 | A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) | ||
Theorem | dchrelbas4 25827* | A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of ℂ, which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑍) ⇒ ⊢ (𝑋 ∈ 𝐷 ↔ (𝑁 ∈ ℕ ∧ 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ ∀𝑥 ∈ ℤ (1 < (𝑥 gcd 𝑁) → (𝑋‘(𝐿‘𝑥)) = 0))) | ||
Theorem | dchrzrh1 25828 | Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘1)) = 1) | ||
Theorem | dchrzrhcl 25829 | A Dirichlet character takes values in the complex numbers. (Contributed by Mario Carneiro, 12-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘𝐴)) ∈ ℂ) | ||
Theorem | dchrzrhmul 25830 | A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐿 = (ℤRHom‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ ℤ) & ⊢ (𝜑 → 𝐶 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) | ||
Theorem | dchrplusg 25831 | Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → · = ( ∘f · ↾ (𝐷 × 𝐷))) | ||
Theorem | dchrmul 25832 | Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 ∘f · 𝑌)) | ||
Theorem | dchrmulcl 25833 | Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝑌 ∈ 𝐷) ⇒ ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐷) | ||
Theorem | dchrn0 25834 | A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝑋‘𝐴) ≠ 0 ↔ 𝐴 ∈ 𝑈)) | ||
Theorem | dchr1cl 25835* | Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ (𝜑 → 𝑁 ∈ ℕ) ⇒ ⊢ (𝜑 → 1 ∈ 𝐷) | ||
Theorem | dchrmulid2 25836* | Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) ⇒ ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋) | ||
Theorem | dchrinvcl 25837* | Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) & ⊢ 𝑍 = (ℤ/nℤ‘𝑁) & ⊢ 𝐷 = (Base‘𝐺) & ⊢ 𝐵 = (Base‘𝑍) & ⊢ 𝑈 = (Unit‘𝑍) & ⊢ 1 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, 1, 0)) & ⊢ · = (+g‘𝐺) & ⊢ (𝜑 → 𝑋 ∈ 𝐷) & ⊢ 𝐾 = (𝑘 ∈ 𝐵 ↦ if(𝑘 ∈ 𝑈, (1 / (𝑋‘𝑘)), 0)) ⇒ ⊢ (𝜑 → (𝐾 ∈ 𝐷 ∧ (𝐾 · 𝑋) = 1 )) | ||
Theorem | dchrabl 25838 | The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝐺 = (DChr‘𝑁) ⇒ ⊢ (𝑁 ∈ ℕ → 𝐺 ∈ Abel) | ||
Theorem | dchrfi 25839 | 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 25840 | 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 25841 | 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 25842* | 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 25843 | 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 25844 | 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 25845 | 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 25846 | 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 25847 | 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 25848* | Lemma for dchrpt 25851. (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 25849* | Lemma for dchrpt 25851. (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 25850* | Lemma for dchrpt 25851. (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 25851* | 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 25852* | 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 25853* | 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 25854* | Lemma for sumdchr 25856. (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 25855 | 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 25856* | 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 25857* | 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 25858* | 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 25859 | Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((2 · 𝑁)C𝑁) = ((!‘(2 · 𝑁)) / ((!‘𝑁) · (!‘𝑁)))) | ||
Theorem | pcbcctr 25860* | 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 25861 | 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 25862 | 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 25863 | 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 25864 | A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) |
⊢ (𝑁 ∈ (ℤ≥‘4) → ((4↑𝑁) / 𝑁) < ((2 · 𝑁)C𝑁)) | ||
Theorem | efexple 25865 | 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 25866* | Lemma for bpos1 25867. (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 25867* | 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 25868 | 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 25869 | 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 25870* | Lemma for bpos 25877. Since the binomial coefficient does not have any primes in the range (2𝑁 / 3, 𝑁] or (2𝑁, +∞) by bposlem2 25869 and prmfac1 16053, 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 25871* | Lemma for bpos 25877. (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 25872* | Lemma for bpos 25877. 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 25873* | Lemma for bpos 25877. 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 25874* | Lemma for bpos 25877. 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 25875 | Lemma for bpos 25877. 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 25876* | Lemma for bpos 25877. 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 25877* | 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 25935. 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 25879 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 25878 | Extend class notation with the Legendre symbol function. |
class /L | ||
Definition | df-lgs 25879* | 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 25880 | {-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 25881 | 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 25882 | 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 25883* | 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 25884* | Lemma for lgsfcl2 25887. (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 25885* | 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 25886* | 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 25887* | The function 𝐹 is closed in integers with absolute value less than 1 (namely {-1, 0, 1}, see zabsle1 25880). (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 25888* | 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 25889* | 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 25890* | 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 25891* | Lemma for lgsval2 25897. (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 25892* | Lemma for lgsval4 25901. (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 25893* | 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 25894 | 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 25895 | The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐴 /L 𝑁) ∈ ℤ) | ||
Theorem | lgsle1 25896 | The Legendre symbol has absolute value less than or equal to 1. Together with lgscl 25895 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 25897 | 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 25898 | 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 25899 | 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 25900 | 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 25935. (Contributed by Mario Carneiro, 4-Feb-2015.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝑃 ∈ (ℙ ∖ {2})) → ((𝐴 /L 𝑃) mod 𝑃) = ((𝐴↑((𝑃 − 1) / 2)) mod 𝑃)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |