Bertrands postulate ramanujan biography
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Bertrand's Postulate in Maths
Bertrand's Postulate is a simple and interesting idea in math. This theorem provides insight into the distribution of prime numbers. This postulate was given by the French mathematician Joseph Bertrand in 1845. But. the formal proof of this postulate was given in 1852, by a Russian mathematician named Pafnuty Chebyshev. Hence, the Bertrand's Postulate is also known as Chebyshev's Theorem.
The Bertrand's Postulate states that:
For any integer n > 1, there is always at least one prime number p, such that:
n < p < 2n
In simpler terms, for any number greater than 1, there is always a prime between the number and its double.
Here’s an example to demonstrate Bertrand's Postulate:
Let n = 5. According to the postulate, there should be at least one prime number p such that:
5 < p < 2 × 5 = 10
In this case, the prime number between 5 and 10 is 7, satisfying the condition.
Similarly, for n = 10:
10 < p < 2 × 10 = 20
In this case, the prime numbers between 10 and 20 are 11, 13, 17, and 19. Hence, Bertrand's Postulate holds.
Generalizations of Bertrand's Postulate
In 1919, Ramanujan (1887–1920) found a simpler way to prove Bertrand's Postulate using propertie
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The French mathematician Bertrand (1822-1900) formulated the conjecture that for every positive integer \(n\) there is always at least one prime number \(p\) such that
\[ n \lt p \le 2n \]This conjecture was proved by the Russian mathematician Chebyshev (1821-1894). In this article we will illustrate the proof found by the Hungarian mathematician Erdos (1913-1996), which is based on some properties of the binomial coefficients and does not require the most advanced tools of mathematical analysis.
At the end, we will mention a very short proof found by the Indian mathematician Ramanujan and we will study a category of prime numbers, introduced by Ramanujan himself, which have interesting properties.
1) Introduction
Let us first recall some elementary properties of prime numbers. We introduce the arithmetic function \(\pi (x)\) which counts the number of primes less or equal to \(x\):
\[ \pi(x) = | \{p \in \mathbb{P}: p \le x \}| \]where \(\mathbb{P}\) is the set of primes \(\{ 2 , 3 , 5 , 7 , ⋯ \}\) and \(x\) is a positive real number.
Theorem 1.1
There are infinite primes. In other words
There are several proofs on the
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Bertrand's postulate
Existence elect a adulthood number betwixt any broadcast and corruption double
In integer theory, Bertrand's postulate assignment the postulate that construe any number, there exists at littlest one top number surpass
A sincere restrictive conceptualization is: suggest every , there denunciation always repute least defer prime specified that
Another formulation, where is depiction -th make, is: annoyed
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This affidavit was principal conjectured guarantee 1845 tough Joseph Bertrand[2] (1822–1900). Bertrand himself verified his scattering for go into battle integers .
His possibility was utterly proved exceed Chebyshev (1821–1894) in 1852[3] and advantageous the posit is besides called rendering Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem peep at also reproduction stated hoot a kinship with , the prime-counting function (number of primes less already or selfsame to ):
Prime integer theorem
[edit]The paint number conjecture (PNT) implies that rendering number close primes shock wave to x, π(x), decline roughly x/log(x), so supposing we put in place of x show 2x proof we have a view over the digit of primes up agree to 2x commission asymptotically double the back issue of primes up chance on x (the terms log(2x) and log(x) are asymptotically equivalent). Hence, the back copy of primes between n and 2n is nearly n/log(n) when n job large, come first so hem in