A formalized proof of Dirichlet's theorem on primes ... - Semantic Scholar
A formalized proof of Dirichlet’s theorem on primes in arithmetic progression JOHN HARRISON Intel Corporation JF1-13, 2111 NE 25th Ave, Hillsboro OR 97124, USA [email protected]
We describe the formalization using the HOL Light theorem prover of Dirichlet’s theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a starting-point. The goal of this paper is twofold. First we describe a simple and efficient proof of the theorem informally, which is otherwise difficult to find in one self-contained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days.
1.
INTRODUCTION
Dirichlet’s theorem asserts that for all pairs of positive integers k and d that are coprime (have no common integer factor besides 1), there are infinitely many primes p such that p ≡ k (mod d), i.e. that the infinite arithmetic progression k, k + d, k + 2d, k + 3d, . . . contains infinitely many primes. (The coprimality condition is easily seen to be necessary, for any common divisor of k and d would divide all members of this progression.) This result was first conjectured by Euler in the case k = 1, and by Legendre in full generality. It was first proved in 1837 by Dirichlet [Dir37], who in the process introduced L-functions and, indeed, more or less began the subject of analytic number theory in its modern form. In this paper, we will present an elementary self-contained proof of Dirichlet’s theorem culled from various sources, and describe its complete formalization in the HOL Light theorem prover [Har96], the culmination of which is the following formal statement: |- ∀d k. 1
We describe the formalization using the HOL Light theorem prover of Dirichlet’s theorem on primes in arithmetic progression. The proof turned out to be more straightforward than expected, but this depended on a careful choice of an informal proof to use as a starting-point. The goal of this paper is twofold. First we describe a simple and efficient proof of the theorem informally, which is otherwise difficult to find in one self-contained place at an elementary level. We also describe its, largely routine, HOL Light formalization, a task that took only a few days.
1.
INTRODUCTION
Dirichlet’s theorem asserts that for all pairs of positive integers k and d that are coprime (have no common integer factor besides 1), there are infinitely many primes p such that p ≡ k (mod d), i.e. that the infinite arithmetic progression k, k + d, k + 2d, k + 3d, . . . contains infinitely many primes. (The coprimality condition is easily seen to be necessary, for any common divisor of k and d would divide all members of this progression.) This result was first conjectured by Euler in the case k = 1, and by Legendre in full generality. It was first proved in 1837 by Dirichlet [Dir37], who in the process introduced L-functions and, indeed, more or less began the subject of analytic number theory in its modern form. In this paper, we will present an elementary self-contained proof of Dirichlet’s theorem culled from various sources, and describe its complete formalization in the HOL Light theorem prover [Har96], the culmination of which is the following formal statement: |- ∀d k. 1
