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In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined closely, the proof is less a statement about topology than a statement about certain properties of arithmetic sequences.
Euclid offered a proof published in his work Elements (Book IX, Proposition 20), [1] which is paraphrased here. [2] Consider any finite list of prime numbers p 1, p 2, ..., p n. It will be shown that there exists at least one additional prime number not included in this list. Let P be the product of all the prime numbers in the list: P = p 1 p ...
Not all Euclid numbers are prime. E 6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number.. Every Euclid number is congruent to 3 modulo 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4.
Both the Furstenberg and Golomb topologies furnish a proof that there are infinitely many prime numbers. [1] [2] A sketch of the proof runs as follows: Fix a prime p and note that the (positive, in the Golomb space case) integers are a union of finitely many residue classes modulo p. Each residue class is an arithmetic progression, and thus clopen.
The proof is due to Ivan Niven, [4] adapted from the product expansion idea of Euler. In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes. The proof rests upon the following four inequalities:
Because the set of primes is a computably enumerable set, by Matiyasevich's theorem, it can be obtained from a system of Diophantine equations. Jones et al. (1976) found an explicit set of 14 Diophantine equations in 26 variables, such that a given number k + 2 is prime if and only if that system has a solution in nonnegative integers: [7]
Although the proof of Dirichlet's Theorem makes use of calculus and analytic number theory, some proofs of examples are much more straightforward. In particular, the proof of the example of infinitely many primes of the form + makes an argument similar to the one made in the proof of Euclid's theorem (Silverman 2013). The proof is given below:
Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges