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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
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.
Copeland and Erdős's proof that their constant is normal relies only on the fact that is strictly increasing and = + (), where is the n th prime number. More generally, if is any strictly increasing sequence of natural numbers such that = + and is any natural number greater than or equal to 2, then the constant obtained by concatenating "0."
Daniel Larsen (born 2003) is an American mathematician known for proving [1] a 1994 conjecture of W. R. Alford, Andrew Granville and Carl Pomerance on the distribution of Carmichael numbers, commonly known as Bertrand's postulate for Carmichael numbers. [2]
The binary conjecture is sufficient for Bertrand's postulate, whereas Bertrand's postulate is necessary for the binary conjecture. Also, the binary conjecture and the ternary conjecture are equivalent. If one is true, so is the other. 2605:E000:6116:7D00:4CD6:5569:EA6F:731C 14:40, 4 October 2017 (UTC)
I believe Erdos' original method puts the bound at 4001 with a more relaxed method. By contrast, the Tochiori bound at 64 works effortlessly and can be done entirely by hand. Moreover, that section is the only translation of Tochiori's paper I am aware of.
A spreading Gaussian distribution of distinct primes illustrating the Erdos-Kac theorem. Around 12.6% of 10,000 digit numbers are constructed from 10 distinct prime numbers and around 68% are constructed from between 7 and 13 primes. A hollow sphere the size of the planet Earth filled with fine sand would have around 10 33 grains.
There are two closely related variants of the Erdős–Rényi random graph model. A graph generated by the binomial model of Erdős and Rényi (p = 0.01)In the (,) model, a graph is chosen uniformly at random from the collection of all graphs which have nodes and edges.