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The philosopher Torkel Franzén, in a book on logic, states, "Euclid's proof that there are infinitely many primes is not an indirect proof [...] The argument is sometimes formulated as an indirect proof by replacing it with the assumption 'Suppose q 1, ..., q n are all the primes'. However, since this assumption isn't even used in the proof ...
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.
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:
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.
It follows immediately that there are infinitely many AP-k for any k. If an AP-k does not begin with the prime k, then the common difference is a multiple of the primorial k# = 2·3·5·...·j, where j is the largest prime ≤ k. Proof: Let the AP-k be a·n + b for k consecutive values of n.
When two primes have a difference of 2, they’re called twin primes. So 11 and 13 are twin primes, as are 599 and 601. Now, it's a Day 1 Number Theory fact that there are infinitely many prime ...
(The list of known primes of this form is A002496.) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as the Bunyakovsky conjecture and Bateman–Horn conjecture. As of 2024, this problem is open. One example of near-square primes are Fermat primes.
In 1954 Carlitz gave a simple proof of the weaker result that there are in general infinitely many irregular primes. [ 7 ] Metsänkylä proved in 1971 that for any integer T > 6 , there are infinitely many irregular primes not of the form mT + 1 or mT − 1 , [ 8 ] and later generalized this.