Bertrand's Postulate Research Articles

On Grzegorczyk induction

We investigate the “mathematical” strength of the theory I∄ ∗ 2 . In particular we prove the quadratic reciprocity law and Bertrand's postulate, using fragments of I∄ ∗ 2 which employ some well-known number-theoretic functions.

On Some Irrational Decimal Fractions

a = 0.235711131719... is irrational, where the sequence of digits is formed by the primes in ascending order. In [1, Th. 138] there are two different proofs for this statement. The first uses a special case of the Dirichlet's theorem, namely: any arithmetical progression of the form 10S 1k + 1 (k = 1,2, . ..) contains primes. In the second proof it is assumed that there is a prime between N and iON for every N > 0, which is the special case of the Bertrand's Postulate. Similar proofs are found in [2]. In this article we will give a direct proof for this statement. We prove even more.

Bertrand's Postulate for primes in arithmetical progressions

Bertrand's Postulate is the theorem that the interval ( x, 2 x) contains at least one prime for x \\2>1. We prove, building on work of Erd\\”os, analogues of this result, in which the interval is of the form ( x, zx) and there are at least m primes ≡ a(mod d) required to be contained in this interval, and where z, a and d have to satisfy some conditions. For the case m  1 the results are worked out using a computer. They can be found in Table 1.

The generalization and proof of Bertrand′s postulate

The purpose of this paper is to show that for 0 < r < 1 one can determine explicitly an x0 such that ∀x ≥ x0, ∃ at least one prime between rx and x. This is a generalization of Bertrand′s Postulate. Furthermore, the same procedures are used to show that if one can find upper and lower bounds for θ(x) whose difference is kxρ then ∃ a prime between x and x − Kxρ, where k, K > 0 are constants, 0 < ρ < 1 and , where p runs over the primes.

The Discriminator (A Simple Application of Bertrand's Postulate)

The Discriminator (A Simple Application of Bertrand's Postulate)

The Discriminator (A Simple Application of Bertrand's Postulate)

The Discriminator (A Simple Application of Bertrand's Postulate)

Calculations for Bertrand's Postulate

In 1845 J. Bertrand conjectured in [1] that for all n > 6, there is at least one prime between n/2 and n 2. Actually, Euler had already formulated this conjecture nearly a hundred years earlier (see [9]). He used. the form which now is well known as Bertrand's postulate, 7r(2 n) -g(n) > 0 for all n > 1, where r( n) denotes the number of primes less than or equal to n. The first proof was given by Tchebychev in [15], 369-382. Moreover, Tchebychev proved the following extension of Bertrand's postulate for e0 = 1/ 5: For any e > eo > 0, there exists an n (e) so that for all n >n(e),

Calculations for Bertrand's Postulate

Calculations for Bertrand's Postulate

Addendum to "A Stronger Bertrand's Postulate with an Application to Partitions"

Addendum to "A Stronger Bertrand's Postulate with an Application to Partitions"

A Stronger Bertrand's Postulate with an Application to Partitions

A Stronger Bertrand's Postulate with an Application to Partitions

On a Problem in Number Theory

Terminology. The sequence of natural numbers begins with 1; 2 is the first prime, but 1 is not a composite number. From Bertrand's postulate that for every n > 1 there is a prime P such that n <P < 2n, it follows that Pn+1 < 2Pn, for all n, where Pn denotes the nth prime. From Nagura's result [2] that for every n ?25 there is a prime P such that n <P ? 6n/5, it follows that Pnu2 < 2Pn for n ? 4. These facts will be used in the proofs of the two lemmas which will be presented before the proof of the theorem itself.