About the formalization of some results by The Chebyshev in number - PowerPoint PPT Presentation

Introduction About the formalization of some results by The Chebyshev in number theory factorization of n ! Upper and lower via the Matita ITP bounds for B Chebishev’s Ψ function Bertrand’s Dipartimento di Scienze dell’Informazione postulate Erd¨ os approach Mura Anteo Zamboni 7, Bologna (1932) Automatic check asperti@cs.unibo.it January 19, 2009

Outline Introduction Introduction 1 The factorization of n ! The factorization of n ! 2 Upper and lower bounds for B Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s Chebishev’s Ψ function postulate 3 Erd¨ os approach (1932) Automatic check 4 Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Matita in a nutshell Introduction The factorization of n ! Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Matita in a nutshell Introduction A light version of Coq. The factorization of n ! Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Matita in a nutshell Introduction A light version of Coq. The factorization of n ! Some distinctive features: Upper and lower bounds for B a primitive notion of metavariable Chebishev’s Ψ function a sophisticated disambiguation mechanism Bertrand’s postulate a powerful coercion system Erd¨ os approach (1932) Automatic check tynicals a mathml compliant goal window semantic selection, cut & paste

Style of the talk Introduction The factorization of n ! Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Style of the talk Introduction The factorization of n ! I will describe the subject in a way suited to formalization Upper and lower but not the formal details. bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Style of the talk Introduction The factorization of n ! I will describe the subject in a way suited to formalization Upper and lower but not the formal details. bounds for B Chebishev’s Ψ function Bertrand’s postulate At a few points I will point out some tricky aspects of the Erd¨ os approach (1932) formal encoding. Automatic check

The Prime Number Theorem Introduction The Let π ( n ) denote the number of primes not exceeding n . factorization of n ! Upper and lower bounds for B Theorem (Hadamard and La Vall´ e Poussin, 1896) Chebishev’s Ψ function π ( n ) ∼ n / log ( n ) Bertrand’s postulate Erd¨ os approach (1932) Automatic check

The Prime Number Theorem Introduction The Let π ( n ) denote the number of primes not exceeding n . factorization of n ! Upper and lower bounds for B Theorem (Hadamard and La Vall´ e Poussin, 1896) Chebishev’s Ψ function π ( n ) ∼ n / log ( n ) Bertrand’s postulate Erd¨ os approach (1932) Formalized by Avigad et al. in Isabelle (ACM-TOCL 9(1), Automatic check 2007), following Selberg’s “elementary” proof (1949).

Chebyshev’s Theorem Theorem (Chebyshev, 1850) Introduction There are two constants c 1 and c 2 such that, for any n The factorization of n n n ! c 1 log ( n ) ≤ π ( n ) ≤ c 2 Upper and lower log ( n ) bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Chebyshev’s Theorem Theorem (Chebyshev, 1850) Introduction There are two constants c 1 and c 2 such that, for any n The factorization of n n n ! c 1 log ( n ) ≤ π ( n ) ≤ c 2 Upper and lower log ( n ) bounds for B Chebishev’s Ψ function Motivations for the formalization: Bertrand’s postulate Erd¨ os approach (1932) Automatic check

Chebyshev’s Theorem Theorem (Chebyshev, 1850) Introduction There are two constants c 1 and c 2 such that, for any n The factorization of n n n ! c 1 log ( n ) ≤ π ( n ) ≤ c 2 Upper and lower log ( n ) bounds for B Chebishev’s Ψ function Motivations for the formalization: Bertrand’s important machinery for number theory: ψ, θ, . . . postulate Erd¨ os approach (1932) Automatic check

Chebyshev’s Theorem Theorem (Chebyshev, 1850) Introduction There are two constants c 1 and c 2 such that, for any n The factorization of n n n ! c 1 log ( n ) ≤ π ( n ) ≤ c 2 Upper and lower log ( n ) bounds for B Chebishev’s Ψ function Motivations for the formalization: Bertrand’s important machinery for number theory: ψ, θ, . . . postulate Erd¨ os approach (1932) methodology: provide a purely arithmetical (and Automatic check constructive) formalization

Chebyshev’s Theorem Theorem (Chebyshev, 1850) Introduction There are two constants c 1 and c 2 such that, for any n The factorization of n n n ! c 1 log ( n ) ≤ π ( n ) ≤ c 2 Upper and lower log ( n ) bounds for B Chebishev’s Ψ function Motivations for the formalization: Bertrand’s important machinery for number theory: ψ, θ, . . . postulate Erd¨ os approach (1932) methodology: provide a purely arithmetical (and Automatic check constructive) formalization To spare logs: 2 c 1 n ≤ n π ( n ) ≤ 2 c 2 n

Outline Introduction Introduction 1 The factorization of n ! The factorization of n ! 2 Upper and lower bounds for B Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s Chebishev’s Ψ function postulate 3 Erd¨ os approach (1932) Automatic check 4 Bertrand’s postulate Erd¨ os approach (1932) Automatic check

The factorization of n ! Introduction Chebyshev’s approach: exploit the decomposition of the The number n ! as a product of prime numbers. factorization of n ! Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

The factorization of n ! Introduction Chebyshev’s approach: exploit the decomposition of the The number n ! as a product of prime numbers. factorization of n ! Upper and lower bounds for B For any prime p , the numbers 1 , 2 , . . . , n include just n Chebishev’s Ψ p function multiples of p , n p 2 multiples of p 2 , an so on. Hence Bertrand’s postulate Erd¨ os approach p n / p i + 1 � � (1932) n ! = (1) Automatic check p ≤ n i < log p ( n ) (see e.g. Hardy & Wright’s, pag. 342).

A formal proof:(1) the factorization of n Introduction Every integer n may be uniquely decomposed as the The factorization of product of all its prime factors. n ! Upper and lower bounds for B Le us write ord p ( n ) for the multiplicity of p in n ; then Chebishev’s Ψ function p ord p ( n ) = � � � Bertrand’s n = p (2) postulate Erd¨ os approach p ≤ n p ≤ n i < log p ( n ) (1932) Automatic check p i + 1 | n for p prime.

A formal proof:(2) the factorization of n A direct proof by induction on the upper bound of the Introduction product. The factorization of n ! Upper and lower bounds for B Chebishev’s Ψ function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

A formal proof:(2) the factorization of n A direct proof by induction on the upper bound of the Introduction product. We have to rephrase the statement in the form The factorization of n ! � p ord p ( n ) Upper and lower ∀ m > c ( n ) , n = bounds for B Chebishev’s Ψ p ≤ m function Bertrand’s postulate Erd¨ os approach (1932) Automatic check

A formal proof:(2) the factorization of n A direct proof by induction on the upper bound of the Introduction product. We have to rephrase the statement in the form The factorization of n ! � p ord p ( n ) Upper and lower ∀ m > c ( n ) , n = bounds for B Chebishev’s Ψ p ≤ m function Bertrand’s To make induction work c ( n ) must be miminal: in this case, postulate Erd¨ os approach the largest prime factor of n ( mpf ( n ) ) (1932) Automatic check � p ord p ( n ) ∀ m > mpf ( n ) , n = p ≤ m

A formal proof:(3) the factorization of n in matita Introduction The ✞ ☎ factorization of definition mpf n := max n ( λ i .primeb i ∧ i | n). n ! Upper and lower bounds for B theorem lt max to pi p primeb: Chebishev’s Ψ function ∀ m,n. Bertrand’s O < n → postulate mpf n < m → Erd¨ os approach (1932) n = pi p m ( λ i .primeb i ∧ i | n) ( λ p.pˆ(ord n p)). Automatic check ✝ ✆

A formal proof:(4) the factorization of n ! Y n ! = m Introduction 1 ≤ m ≤ n The factorization of Y Y Y = p n ! Upper and lower 1 ≤ m ≤ n p ≤ m i < log p ( m ) bounds for B p i + 1 | m Chebishev’s Ψ function Y Y Y = p Bertrand’s postulate p ≤ n p ≤ m ≤ n i < log p ( m ) Erd¨ os approach p i + 1 | m (1932) Automatic check Y Y Y = p p ≤ n i < log p ( n ) m ≤ n p i + 1 | m p n / p i + 1 Y Y = p ≤ n i < log p ( n )

The binomial coefficient B ( 2 n ) = ( 2 n n ) For 2 n we have: p 2 n / p i + 1 Y Y 2 n ! = (3) Introduction p ≤ 2 n i < log p ( 2 n ) The But factorization of p i + 1 = 2 n 2 n p i + 1 + ( 2 n n ! p i + 1 mod 2 ) Upper and lower bounds for B Moreover, if n ≤ p or log p ( n ) ≤ i we have Chebishev’s Ψ function n p i + 1 = O Bertrand’s postulate Erd¨ os approach (1932) Hence, if we define Automatic check p ( n / p i + 1 mod 2 ) Y Y B ( n ) = p ≤ n i < log p ( n ) equation (3) becomes 2 n ! = n ! 2 B ( 2 n ) (4)