Bertrand's Postulate Research Articles

Sharing proofs with predicative theories through universe-polymorphic elaboration

As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism. It consists in trying to elaborate each term into a predicative universe-polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the elaboration, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in a partial algorithm which uses a constraint-postponement strategy for trying to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's library to Agda, including proofs of Bertrand's Postulate and Fermat's Little Theorem, which (as far as we know) were not available in Agda yet.

Confirming Buya's and Bezaleel's Proof of the Binary Goldbach Conjecture using Bertrand's Postulate Paper Title

Confirming Buya's and Bezaleel's Proof of the Binary Goldbach Conjecture using Bertrand's Postulate Paper Title

An Exploration of Very Triangular Numbers

We present a collection of results concerning the location and distribution of very triangular numbers among triangular numbers, including the twin very triangular number theorem, the existence of arbitrarily long gaps between – and an analog of Bertrand's postulate for – very triangular numbers.

An Inductive Proof of Bertrand's Postulate

In this paper, we are going to prove a famous problem concerning the prime numbers called Bertrand's postulate. It states that there is always at least one prime, p between n and 2n, means, there exists n < p < 2n where n > 1. It is not a newer theorem to be proven. It was first conjectured by Joseph Bertrand in 1845. He did not find a proof of this problem but made important numerical evidence for the large values of n. Eventually, it was successfully proven by Pafnuty Chebyshev in 1852. That is why it is also called Bertrand-Chebyshev theorem. Though it does not give very strong idea about the prime distribution like Prime Number Theorem (PNT) does, the beauty of Bertrand's postulate lies on its simple yet elegant definition. Historically, Bertrand's postulate is also very important. After Euclid's proof that there are infinite prime numbers, there was no significant development in the prime number distribution. Peter Dirichlet stated the standard form of Prime Number Theorem (PNT) in 1838 but it was merely a conjecture that time and beyond the scope of proof to the then mathematicians. Bertrand's postulate was a simply stated problem but powerful enough, easy to prove and could lead many more strong assumptions about the prime number distribution. Illustrious Indian mathematician, Srinivasa Ramanujan gave a shorter but elegant proof using the concept of Chebyshev functions of prime, υ(x), Ψ(x)and Gamma function, Γ(x) in 1919 which led to the concept of Ramanujan Prime. Later Paul Erdős published another proof using the concept of Primorial function, p# in 1932. The elegance of our proof lies on not using Gamma function yet finding the better approximations of Chebyshev functions of prime. The proof technique is very similar the way Ramanujan proved it but instead of using the Stirling's approximation to the binomial coefficients, we are proving similar results using well-known proving technique the mathematical induction and they lead to somewhat stronger than Ramanujan's approximation of Chebyshev functions of prime.
 GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 85-87

Bertrand’s postulate for number fields

Consider an algebraic number field, $K$, and its ring of integers, $\mathcal{O}_K$. There exists a smallest $B_K>1$ such that for any $x>1$ we can find a prime ideal, $\mathfrak{p}$, in $\mathcal{O}_K$ with norm $N(\mathfrak{p})$ in the interval $[x,B_Kx]$. This is a generalization of Bertrand's postulate to number fields, and in this paper we produce bounds on $B_K$ in terms of the invariants of $K$ from an effective prime ideal theorem due to Lagarias and Odlyzko. We also show that a bound on $B_K$ can be obtained from an asymptotic estimate for the number of ideals in $\mathcal{O}_K$ less than $x$.

A proof of Bertrand's postulate

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate. Even if Chebyshev's result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions psi and theta still play a central role in the modern development of number theory. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base.

Schatunowsky's theorem, Bonse's inequality, and Chebyshev's theorem in weak fragments of Peano arithmetic

In 1893, Schatunowsky showed that 30 is the largest number all of whose totatives are primes; we show that this result cannot be proved, in any form, in Chebyshev's theorem (Bertrand's postulate), even if all irreducibles are primes. Bonse's inequality is shown to be indeed weaker than Chebyshev's theorem. Schatunowsky's theorem holds in together with Bonse's inequality, the existence of the greatest prime dividing certain types of numbers, and the condition that all irreducibles be prime.

On the Function π(x)

Abstract Let π(x) be the number of primes not exceeding x. We prove that π ( x ) < x log x - 1.006789 for x ≥ e 10 12 , and that for sufficiently large x : x log x - 1 + ( log x ) - 1.5 + 2 ( log x ) - 0.5 < π ( x ) < 1 log x - 1 - 2 ( log x ) - 0.5 - ( log x ) - 1.5 . We finally prove that for x ≥ e 10 12 and k = 2, 3,…, 147297098200000, the closed interval [(k – 1)x, kx] contains at least one prime number, i.e. the Bertrand's postulate holds for x and k as above.

Ramanujan's Proof of Bertrand's Postulate

We present Ramanujan's proof of Bertrand's postulate and in the process, eliminate his use of Stirling's formula. The revised proof is elegant and elementary so as to be accessible to a wider audience.

Are the hyperharmonics integral? A partial answer via the small intervals containing primes

In a recent work, the authors have used Bertrand's postulate to give a partial answer to the conjecture of Mező which says that the hyperharmonic numbers – iterations of partial sums of harmonic numbers – are not integers. In this Note, using small intervals containing prime numbers, we prove that a great class of hyperharmonic numbers are not integers.

Bertrand's postulate and subgroup growth

In this article we investigate the L^1-norm of certain functions on groups called divisibility functions. Using these functions, their connection to residual finiteness, and integration theory on profinite groups, we define the residual average of a finitely generated group. One of the main results in this article is the finiteness of residual averages on finitely generated linear groups. Whether or not the residual average is finite depends on growth rates of indices of finite index subgroups. Our results on index growth rates are analogous to results on gaps between primes, and provide a variant of the subgroup growth function, which may be of independent interest.

Ramanujan Primes and Bertrand's Postulate

(2009). Ramanujan Primes and Bertrand's Postulate. The American Mathematical Monthly: Vol. 116, No. 7, pp. 630-635.

Ramanujan Primes and Bertrand's Postulate

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and gave the first five values as 2, 11, 17, 29, 41. In this note, we use inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log 4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is asymptotic to the $2n$th prime. We also estimate the length of the longest string of consecutive Ramanujan primes among the first $n$ primes, explain why there are more twin Ramanujan primes than expected, and make three conjectures (the first has since been proved by S. Laishram).

Bertrand's postulate, the prime number theorem and product anti-magic graphs

Let the edges of a finite simple graph G = ( V , E ) , | V | = n , | E | = m , be labeled by 1 , 2 , … , m . Denote by w ( u ) the product of all the labels of edges incident with a vertex u . The graph G is called product anti-magic if it is possible that the above labeling results in all values w ( u ) being distinct. Following an old conjecture of Hartsfield and Ringel on (sum) anti-magic graphs (see [N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Inc., Boston, 1990, pp. 108–109 (revised version, 1994)]), Figueroa-Centeno et al. [Bertrand's postulate and magical product labellings, Bull. ICA 30 (2000) 53–65] conjectured that every connected graph of size m is product anti-magic iff m ⩾ 3 . In this paper we prove this conjecture for dense graphs, complete multi-partite graphs and some other families of graphs.

Pocklington's Theorem and Bertrand's Postulate

Pocklington's Theorem and Bertrand's Postulate The first four sections of this article include some auxiliary theorems related to number and finite sequence of numbers, in particular a primality test, the Pocklington's theorem (see [19]). The last section presents the formalization of Bertrand's postulate closely following the book [1], pp. 7-9.

Euclid's Argument on the Infinitude of Primes

Of course, better estimates can be obtained from Bertrand's Postulate, or from versions of the Prime Number Theorem having explicit constants, but the proofs of these results are also much harder. Euclid's argument yields some special cases of Dirichlet's great theorem on primes in arithmetic progressions. The preceding modification can also be adapted to this context. In what follows, we use pi to denote the ith prime of the form p = 4k + 3.

Euclid's Argument on the Infinitude of Primes

Of course, better estimates can be obtained from Bertrand's Postulate, or from versions of the Prime Number Theorem having explicit constants, but the proofs of these results are also much harder. Euclid's argument yields some special cases of Dirichlet's great theorem on primes in arithmetic progressions. The preceding modification can also be adapted to this context. In what follows, we use pi to denote the ith prime of the form p = 4k + 3.

Diophantine conditions in small divisors and transcendental number theory

We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.

A course in computational number theory

Preface. Notation. Chapter 1 Fundamentals. 1.0 Introduction. 1.1 A Famous Sequence of Numbers. 1.2 The Euclidean ALgorithm. The Oldest Algorithm. Reversing the Euclidean Algorithm. The Extended GCD Algorithm. The Fundamental Theorem of Arithmetic. Two Applications. 1.3 Modular Arithmetic. 1.4 Fast Powers. A Fast Alforithm for ExponentiationPowers of Matrices, Big-O Notation. Chapter 2 Congruences, Equations, and Powers. 2.0 Introduction. 2.1 Solving Linear Congruences. Linear Diophantine Equations in Two Variables. The Conductor. An Importatnt Quadratic Congruence. 2.2 The Chinese Remainder Theorem. 2.3 PowerMod Patterns. Fermat's Little Theorem. More Patterns in Powers. 2.4 Pseudoprimes. Using the Pseudoprime Test. Chapter 3 Euler's Function. 3.0 Introduction. 3.1 Euler's Function. 3.2 Perfect Numbers and Their Relatives. The Sum of Divisors Function. Perfect Numbers. Amicalbe, Abundant, and Deficient Numbers. 3.3 Euler's Theorem. 3.4 Primitive Roots for Primes. The order of an Integer. Primes Have PRimitive roots. Repeating Decimals. 3.5 Primitive Roots for COmposites. 3.6 The Universal Exponent. Universal Exponents. Power Towers. The Form of Carmichael Numbers. Chapter 4 Prime Numbers. 4.0 Introduction. 4.1 The Number of Primes. We'll Never Run Out of Primes. The Sieve of Eratosthenes. Chebyshev's Theorem and Bertrand's Postulate. 4.2 Prime Testing and Certification. Strong Pseudoprimes. Industrial-Grade Primes. Prime Certification Via Primitive Roots. An Improvement. Pratt Certificates. 4.3 Refinements and Other Directions. Other PRimality Tests. Strong Liars are Scarce. Finding the nth Prime. 4.4 A Doszen Prime Mysteries. Chapter 5 Some Applications. 5.0 Introduction. 5.1 Coding Secrets. Tossing a Coin into a Well. The RSA Cryptosystem. Digital Signatures. 5.2 The Yao Millionaire Problem. 5.3 Check Digits. Basic Check Digit Schemes. A Perfect Check Digit Method. Beyond Perfection: Correcting Errors. 5.4 Factoring Algorithms. Trial Division. Fermat's Algorithm. Pollard Rho. Pollard p-1. The Current Scene. Chapter 6 Quadratic Residues. 6.0 Introduction. 6.1 Pepin's Test. Quadratic Residues. Pepin's Test. Primes Congruent to 1 (Mod. 6.2 Proof of Quadratic Reciprocity. Gauss's Lemma. Proof of Quadratic Recipocity. Jacobi's Extension. An Application to Factoring. 6.3 Quadratic Equations. Chapter 7 Continuec Faction. 7.0 Introduction. 7.1 FInite COntinued Fractions. 7.2 Infinite Continued Fractions. 7.3 Periodic Continued Fractions. 7.4 Pell's Equation. 7.5 Archimedes and the Sun God's Cattle. Wurm's Version: Using Rectangular Bulls. The Real Cattle Problem. 7.6 Factoring via Continued Fractions. Chapter 8 Prime Testing with Lucas Sequences. 8.0 Introduction. 8.1 Divisibility Properties of Lucas Sequencese. 8.2 Prime Tests Using Lucas Sequencesse. Lucas Certification. The Lucas-Lehmer Algorithm Explained. Luca Pseudoprimes. Strong Quadratic Pseudoprimes. Primality Testing's Holy Grail. Chapter 9 Prime Imaginaries and Imaginary Primes. 9.0 Introduction. 9.1 Sums of Two Squares. 9.2 The Gaussian Intergers. Complex Number Theory. Gaussian Primes. The Moat Problem. The Gaussian Zoo. 9.3 Higher Reciprocity 325. Appendix A. Maathematica Basics. 1.0 Introduction. A.1 Plotting. A.2 Typesetting. Sending Files By E-Mail. A.3 Types of Functions. A.4 Lists. A.5 Programs. A.6 Solving Equations. A.7 Symbolic Algebra. Appendix B Lucas Certificates Exist. References. Index of Mathematica Objects. Subject Index.

Rational Points on a Class of Superelliptic Curves

A famous Diophantine equation is given byformula hereFor integers k[ges ]2 and m[ges ]2, this equation only has the solutions x = −j (j = 1, …, m), y = 0 by a remarkable result of Erdős and Selfridge [9] in 1975. This put an end to the old question of whether the product of consecutive positive integers could ever be a perfect power (except for the obviously trivial cases). In a letter to D. Bernoulli in 1724, Goldbach (see [7, p. 679]) showed that (1) has no solution with x[ges ]0 in the case k = 2 and m = 3. In 1857, Liouville [18] derived from Bertrand's postulate that for general k[ges ]2 and m[ges ]2, there is no solution with x[ges ]0 if one of the factors on the right-hand side of (1) is prime. By use of the Thue–Siegel theorem, Erdős and Siegel [10] proved in 1940 that (1) has only trivial solutions for all sufficiently large k[ges ]k0 and all m. This was closely related to Siegel's earlier result [30] from 1929 that the superelliptic equationformula herehas at most finitely many integer solutions x, y under appropriate conditions on the polynomial f(x). The ineffectiveness of k0 was overcome by Baker's method [1] in 1969 (see also [2]).In 1955, Erdős [8] managed to re-prove the result jointly obtained with Siegel by elementary methods. A refinement of Erdős' ideas finally led to the above-mentioned theorem as follows.