A Question on a Wikipedia article on Bertrand's theorem

In summary, the mistake in the derivation of Bertrand's theorem is assuming that β is a constant for all radii, when in fact it varies with each radius.
  • #1
Kashmir
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Wikipedia while deriving Bertrands theorem writes after some steps :
(...For the orbits to be closed, ##β## must be a rational number. What's more, it must be the same rational number for all radii, since ##β## cannot change continuously.Using the definition of ##J## along with equation (1) we get##J^{\prime}\left(u_{0}\right)=\frac{2}{u_{0}}\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right]-\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right] \frac{1}{f\left(\frac{1}{u_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=-2+\frac{u_{0}}{f\left(\frac{1}{v_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=1-\beta^{2}##where ##J^{\prime}\left(u_{0}\right)=\left.\frac{d J}{d u}\right|_{u_{0}}##Since this must hold for any value of ##u_0## then##{\displaystyle {\frac {df}{dr}}=(\beta ^{2}-3){\frac {f}{r}}}## and hence##{\displaystyle f(r)=-{\frac {k}{r^{3-\beta ^{2}}}}.}##and then we find that##{\displaystyle J(u)={\frac {mk}{L^{2}}}u^{1-\beta ^{2}}.}##
However by a similar argument we can say that since ##β## is a constant then we can directly solve
##J^{\prime}(u)=1-\beta^{2}## and find that ##J=\left(1-β^{2}\right) u##) which is wrong.

What went wrong?

 
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  • #2
The mistake in the derivation is that the author mistakenly assumed that the rational number β is the same for all radii. This is incorrect, as Bertrand's theorem states that for each different radius there is a different rational number β associated with it. Thus, the equation J'(u) = 1 - β2 does not hold for any value of u, since the value of β varies depending on u. Therefore, the equation J(u) = (1 - β2)u is incorrect.
 

Related to A Question on a Wikipedia article on Bertrand's theorem

1. What is Bertrand's theorem?

Bertrand's theorem, also known as Bertrand's postulate, is a mathematical theorem that states that for any positive integer n greater than 1, there exists at least one prime number between n and 2n. This theorem was first stated by French mathematician Joseph Bertrand in 1845.

2. What is the significance of Bertrand's theorem?

Bertrand's theorem has significant implications in number theory and has been used to prove other theorems, such as Chebyshev's theorem, which states that there is always at least one prime number between n and (2-ε)n for any positive real number ε. It also has applications in probability theory and has been used to solve various mathematical problems.

3. How was Bertrand's theorem proven?

Bertrand's theorem was first proven by Russian mathematician Pafnuty Chebyshev in 1852. Since then, there have been multiple proofs of the theorem, including one by Paul Erdős in 1932. The most well-known proof is the one by Erdős, which uses the prime number theorem and the Chebyshev function.

4. Is Bertrand's theorem always true?

No, Bertrand's theorem is not always true. It is only true for positive integers greater than 1. For example, there is no prime number between 0 and 2, so the theorem does not hold for n = 1. Additionally, there are some values of n for which the theorem has not been proven, such as n = 10^18.

5. What are some real-world applications of Bertrand's theorem?

Bertrand's theorem has applications in various fields, including cryptography, computer science, and economics. It is also used in the analysis of algorithms and has been used to optimize the performance of computer programs. In economics, the theorem is used to analyze the distribution of wealth and income in a society.

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