Bertrand's Postulate and Erdős' Proof

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Thread starter Gear300
Start date May 2, 2017
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In summary, Bertrand's Postulate, also known as Bertrand's Conjecture, is a mathematical theory that states that for any positive integer n, there exists at least one prime number between n and 2n. It was proven by Hungarian mathematician Paul Erdős and Norwegian mathematician Atle Selberg in 1948 using a combination of combinatorial and analytic methods. This theorem has important implications in number theory and has been used in the development of other mathematical theories and proofs. However, it does have some limitations, as it does not provide information about the distribution or density of prime numbers and cannot prove the existence of infinitely many primes.

May 2, 2017

Gear300

Hello.
Is there a quick proof for showing that the next prime is within twice the current prime?

Edit:
Never mind. Erdős had given a proof of this (of Bertrand's postulate to be precise) at a fairly young age.

http://www3.nd.edu/~dgalvin1/pdf/bertrand.pdf

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May 2, 2017

jedishrfu

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Thanks for sharing this.

1. What is Bertrand's Postulate?

Bertrand's Postulate, also known as Bertrand's Conjecture, is a mathematical theory proposed by French mathematician Joseph Bertrand in 1845. It states that for any positive integer n, there exists at least one prime number between n and 2n.

2. What is Erdős' Proof?

Erdős' Proof, also known as the Erdős-Selberg Proof, is a mathematical proof of Bertrand's Postulate. It was published in 1948 by Hungarian mathematician Paul Erdős and Norwegian mathematician Atle Selberg. The proof uses a combination of combinatorial and analytic methods to show that there is always at least one prime number between n and 2n.

3. Why is Bertrand's Postulate important?

Bertrand's Postulate is important because it provides a key insight into the distribution of prime numbers. It also has many applications in number theory and has been used in the development of other mathematical theories and proofs.

4. Are there any limitations to Bertrand's Postulate?

Yes, there are some limitations to Bertrand's Postulate. While it guarantees the existence of at least one prime number between n and 2n, it does not provide any information about the distribution or density of prime numbers. It also cannot be used to prove the existence of infinitely many primes, which is another famous unsolved problem in mathematics.

5. Has Bertrand's Postulate been proven?

Yes, Bertrand's Postulate has been proven by Erdős and Selberg in 1948. However, there have been subsequent refinements and generalizations of the proof by other mathematicians. It is now considered a well-established theorem in mathematics.