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Substituting these values into the last equation yields the main result of Bertrand's theorem: β 2 ( 1 − β 2 ) ( 4 − β 2 ) = 0. {\displaystyle \beta ^{2}(1-\beta ^{2})(4-\beta ^{2})=0.} Hence, the only potentials that can produce stable closed non-circular orbits are the inverse-square force law ( β = 1 {\displaystyle \beta =1} ) and ...
In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <.First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.
In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method.
In number theory, Bertrand's postulate is the theorem that for any integer >, there exists at least one prime number with n < p < 2 n − 2. {\displaystyle n<p<2n-2.} A less restrictive formulation is: for every n > 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that
They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem). [1]: 92 The Kepler problem also conserves the Laplace–Runge–Lenz vector, which has since been generalized to include other interactions.
Bertrand's box paradox – Mathematical paradox; Bertrand's postulate – Existence of a prime number between any number and its double; Bertrand's theorem – Physics theorem; Bertrand's ballot theorem – Theorem that gives the probability that an election winner will lead the loser throughout the count; Bertrand–Edgeworth model ...
Retrieved from "https://en.wikipedia.org/w/index.php?title=Bertrand_theorem&oldid=111268896"This page was last edited on 27 February 2007, at 05:24 (UTC). (UTC).
Chebyshev's theorem is any of several theorems proven by Russian mathematician Pafnuty Chebyshev. Bertrand's postulate, that for every n there is a prime between n and 2n. Chebyshev's inequality, on the range of standard deviations around the mean, in statistics; Chebyshev's sum inequality, about sums and products of decreasing sequences