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Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane. Thus, as Gaussian primes extend over the plane and not only along a line, and doubling a complex number is not simply multiplying by 2 but doubling its norm (multiplying by 1+i), different ...
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."
Proof of Bertrand's postulate; Fermat's theorem on sums of two squares; Two proofs of the Law of quadratic reciprocity; Proof of Wedderburn's little theorem asserting that every finite division ring is a field; Four proofs of the Basel problem; Proof that e is irrational (also showing the irrationality of certain related numbers) Hilbert's ...
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)
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; Chebyshev's equioscillation theorem, on the approximation of continuous functions with polynomials
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.