This document presents two theorems about repunit Lehmer numbers. Theorem 1 states that for any fixed base g > 1, there are only finitely many positive integers n such that the repunit number un = (gn - 1)/(g - 1) is a Lehmer number, and these can all be effectively computed. Theorem 2 states that there are no Lehmer numbers of the form un when 2 ≤ g ≤ 1000. The document provides background on Lehmer numbers and repunit numbers, establishes some preliminary results, and gives the proof of Theorem 1 by considering primitive divisors of the repunit numbers.
This document discusses predicates and quantifiers in predicate logic. It defines predicates as statements involving variables whose truth depends on variable values. Predicates become propositions when variables are assigned values. Quantifiers like universal (∀) and existential (∃) are used to express the extent to which a predicate is true. Universal quantifiers mean a predicate is true for all variables, while existential quantifiers mean a predicate is true for at least one variable. Examples show how to represent English language sentences using predicates and quantifiers in predicate logic.
The document discusses mathematical induction and recursion.
It defines mathematical induction as a method of proof that establishes a statement is true for all natural numbers by proving the base case and inductive step. Recursion is defined as a programming strategy that solves large problems by splitting them into smaller subproblems of the same kind. Examples are provided to demonstrate proofs by mathematical induction and defining functions recursively.
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
The document describes the principle of mathematical induction and how it is used to prove statements about natural numbers. It contains the following key points:
1. Mathematical induction has two steps: the basis step proves the statement is true for the first natural number, and the inductive step assumes the statement is true for some natural number n, and uses this to prove it is true for n+1.
2. Examples are provided to demonstrate how to apply mathematical induction to prove formulas and inequalities for all natural numbers. The examples show setting up the basis and inductive steps, and using the induction hypothesis within the proof.
3. Inequalities can sometimes be proved using induction by adding or subtracting quantities to both
This document discusses predicates and quantifiers in predicate logic. It defines predicates as statements involving variables whose truth depends on variable values. Predicates become propositions when variables are assigned values. Quantifiers like universal (∀) and existential (∃) are used to express the extent to which a predicate is true. Universal quantifiers mean a predicate is true for all variables, while existential quantifiers mean a predicate is true for at least one variable. Examples show how to represent English language sentences using predicates and quantifiers in predicate logic.
The document discusses mathematical induction and recursion.
It defines mathematical induction as a method of proof that establishes a statement is true for all natural numbers by proving the base case and inductive step. Recursion is defined as a programming strategy that solves large problems by splitting them into smaller subproblems of the same kind. Examples are provided to demonstrate proofs by mathematical induction and defining functions recursively.
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)Yandex
We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.
Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one $k$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).
We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.
References
[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.
1: 97–115, 1988.
The document describes the principle of mathematical induction and how it is used to prove statements about natural numbers. It contains the following key points:
1. Mathematical induction has two steps: the basis step proves the statement is true for the first natural number, and the inductive step assumes the statement is true for some natural number n, and uses this to prove it is true for n+1.
2. Examples are provided to demonstrate how to apply mathematical induction to prove formulas and inequalities for all natural numbers. The examples show setting up the basis and inductive steps, and using the induction hypothesis within the proof.
3. Inequalities can sometimes be proved using induction by adding or subtracting quantities to both
1. The document discusses using the binomial expansion and Stirling's formula to estimate the value of r that maximizes a binomial coefficient expression as n becomes large. Taking the limit as n approaches infinity, the optimal value of r is shown to be nq.
2. A example is given of estimating the probability of winning $40 or more by betting $1 on number 8 in roulette 500 times. Using the normal approximation, this probability is estimated to be about 25.8%.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
Amirim Project - Threshold Functions in Random Simplicial Complexes - Avichai...Avichai Cohen
This document presents an overview of prior work on threshold functions in random simplicial complexes. It discusses questions about the threshold for the existence of cycles and collapsibility in higher dimensional simplicial complexes. Specifically, it summarizes previous results that established coarse threshold functions of 1/n for the existence of cycles in dimensions greater than 1. It also reviews upper bounds on the critical probability for this threshold. The document outlines prior studies on the collapsability threshold and presents lower bounds on the critical probability. The author's own work is aimed at improving the bounds and determining sharp threshold functions for these properties in random simplicial complexes.
[1] Mathematical induction can be used to prove statements about positive integers n, where P(n) is some proposition. It has two parts: the base step verifies P(1) is true, and the inductive step shows that if P(k) is true then P(k+1) is also true.
[2] To prove a statement for all n using induction, you must show P(0) is true and that P(k) implies P(k+1). You can think of it as a recursive proof where proving P(n+1) relies on having already proven P(n).
[3] As an example, the document proves the statement ∑
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
The document discusses using k-nearest neighbors and KD-trees to create a computationally cheap approximation (πa) of an expensive-to-evaluate target distribution π. This approximation allows the use of delayed acceptance in a Metropolis-Hastings or pseudo-marginal Metropolis-Hastings algorithm to potentially reduce computation cost per iteration. Specifically, it describes:
1) Using a weighted average of the k nearest neighbor π values to define the approximation πa.
2) How delayed acceptance preserves the stationary distribution while mixing more slowly than standard MH.
3) Storing the evaluated π values in a KD-tree to enable fast lookup of the k nearest neighbors.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document discusses L1-convergence of the Rees-Stanojević modified cosine sum. It contains the following key points:
1. The author obtains a necessary and sufficient condition for L1-convergence of the Rees-Stanojević sum under the condition that the Fourier coefficients satisfy a certain limit.
2. This generalizes a previous result by Garrett and Stanojević which proved convergence in the L-metric for quasi-convex sequences.
3. As a corollary, the author shows that the L1-convergence of the partial sums of the cosine series is equivalent to the condition that the product of the largest neglected coefficient and log n approaches 0 as n approaches
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document discusses the visual elements of images, identifying dots, lines, planes, color, and texture as the basic building blocks. It provides details on each element, including that a dot is the smallest and simplest, a line is a dot in motion, and planes are larger flat shapes. These elements can be used alone or combined to create images and convey ideas through visual expression.
1. The document discusses using the binomial expansion and Stirling's formula to estimate the value of r that maximizes a binomial coefficient expression as n becomes large. Taking the limit as n approaches infinity, the optimal value of r is shown to be nq.
2. A example is given of estimating the probability of winning $40 or more by betting $1 on number 8 in roulette 500 times. Using the normal approximation, this probability is estimated to be about 25.8%.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
Algorithm Design and Complexity - Course 3Traian Rebedea
The document provides an overview of recursive algorithms and complexity analysis. It discusses recursive algorithms, divide and conquer design technique, and several examples of recursive algorithms including Towers of Hanoi, Merge Sort, and Quick Sort. For recursive algorithms, it explains how to analyze their running time using recurrence relations. It then covers four methods for solving recurrence relations: iteration, recursion trees, substitution method, and master theorem. The substitution method and master theorem are described as the most rigorous mathematical approaches.
Amirim Project - Threshold Functions in Random Simplicial Complexes - Avichai...Avichai Cohen
This document presents an overview of prior work on threshold functions in random simplicial complexes. It discusses questions about the threshold for the existence of cycles and collapsibility in higher dimensional simplicial complexes. Specifically, it summarizes previous results that established coarse threshold functions of 1/n for the existence of cycles in dimensions greater than 1. It also reviews upper bounds on the critical probability for this threshold. The document outlines prior studies on the collapsability threshold and presents lower bounds on the critical probability. The author's own work is aimed at improving the bounds and determining sharp threshold functions for these properties in random simplicial complexes.
[1] Mathematical induction can be used to prove statements about positive integers n, where P(n) is some proposition. It has two parts: the base step verifies P(1) is true, and the inductive step shows that if P(k) is true then P(k+1) is also true.
[2] To prove a statement for all n using induction, you must show P(0) is true and that P(k) implies P(k+1). You can think of it as a recursive proof where proving P(n+1) relies on having already proven P(n).
[3] As an example, the document proves the statement ∑
Density theorems for Euclidean point configurationsVjekoslavKovac1
1. The document discusses density theorems for point configurations in Euclidean space. Density theorems study when a measurable set A contained in Euclidean space can be considered "large".
2. One classical result is that for any measurable set A contained in R2 with positive upper Banach density, there exist points in A whose distance is any sufficiently large real number. This has been generalized to higher dimensions and other point configurations.
3. Open questions remain about determining all point configurations P for which one can show that a sufficiently large measurable set A contained in high dimensional Euclidean space must contain a scaled copy of P.
The document discusses scaling sets and MRA wavelet sets, which are measurable sets associated with multiresolution analyses and wavelets. It provides definitions and theorems characterizing scaling sets and MRA wavelet sets. Some simple examples of scaling sets and MRA wavelet sets are given as finite unions of intervals. The document then poses questions about the properties of general wavelet sets and provides counterexamples to ideas about possible restrictions on their structure. Finally, more complex examples of scaling sets and MRA wavelet sets are constructed using Rademacher functions.
A T(1)-type theorem for entangled multilinear Calderon-Zygmund operatorsVjekoslavKovac1
This document summarizes a talk given by Vjekoslav Kovač at a joint mathematics conference. The talk concerned establishing T(1)-type theorems for entangled multilinear Calderón-Zygmund operators. Specifically, Kovač discussed studying multilinear singular integral forms where the functions partially share variables, known as an "entangled structure." He outlined establishing generalized modulation invariance and Lp estimates for such operators. The talk motivated further studying related problems involving bilinear ergodic averages and forms with more complex graph structures. Kovač specialized his techniques to bipartite graphs, multilinear Calderón-Zygmund kernels, and "perfect" dyadic models.
Tales on two commuting transformations or flowsVjekoslavKovac1
1) The document summarizes recent work on ergodic averages and flows for commuting transformations. It discusses convergence results for single and double linear ergodic averages in L2 and almost everywhere, as well as providing norm estimates to quantify the rate of convergence.
2) It also considers double polynomial ergodic averages and provides proofs for almost everywhere convergence in the continuous-time setting. Open problems remain for the discrete-time case.
3) An ergodic-martingale paraproduct is introduced, motivated by an open question from 1950. Convergence in Lp norm is shown, while almost everywhere convergence remains open.
The document discusses using k-nearest neighbors and KD-trees to create a computationally cheap approximation (πa) of an expensive-to-evaluate target distribution π. This approximation allows the use of delayed acceptance in a Metropolis-Hastings or pseudo-marginal Metropolis-Hastings algorithm to potentially reduce computation cost per iteration. Specifically, it describes:
1) Using a weighted average of the k nearest neighbor π values to define the approximation πa.
2) How delayed acceptance preserves the stationary distribution while mixing more slowly than standard MH.
3) Storing the evaluated π values in a KD-tree to enable fast lookup of the k nearest neighbors.
Trilinear embedding for divergence-form operatorsVjekoslavKovac1
The document discusses a trilinear embedding theorem for divergence-form operators with complex coefficients. It proves that if matrices A, B, C are appropriately p,q,r-elliptic, then there is a bound on the integral of the product of the gradients of the semigroups associated with the operators. The proof uses a Bellman function technique and shows the relationship to the concept of p-ellipticity. It generalizes previous work on bilinear embeddings to the trilinear case.
Scattering theory analogues of several classical estimates in Fourier analysisVjekoslavKovac1
This document summarizes some classical estimates in Fourier analysis and their analogues in nonlinear Fourier analysis. It discusses Carleson's theorem on convergence of Fourier series and Fourier transforms, Hausdorff-Young inequalities bounding Lp norms, and results on lacunary trigonometric series and products. Open questions are presented about extending these classical estimates to the nonlinear setting of the SU(1,1) Fourier transform and lacunary SU(1,1) trigonometric products.
Variants of the Christ-Kiselev lemma and an application to the maximal Fourie...VjekoslavKovac1
1. The document discusses variants of the Christ-Kiselev lemma and its application to maximal Fourier restriction estimates.
2. The Christ-Kiselev lemma allows block-diagonal and block-triangular truncations of operators while controlling their operator norms.
3. These lemmas can be used to prove maximal and variational estimates for the restriction of the Fourier transform to surfaces, which has applications in harmonic analysis.
This document summarizes research on norm-variation estimates for ergodic bilinear and multiple averages. It begins by motivating the study of ergodic averages and their convergence properties. Previous results are discussed that provide pointwise convergence and norm estimates for certain cases. The document then presents new norm-variation estimates obtained by the authors for bilinear and multiple ergodic averages over general measure-preserving systems. These estimates bound the number of jumps in the L2 norm as the averages converge. Finally, analogous results are discussed for bilinear averages on R2 and Z2, linking the estimates to established bounds for singular integrals.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
The document discusses L1-convergence of the Rees-Stanojević modified cosine sum. It contains the following key points:
1. The author obtains a necessary and sufficient condition for L1-convergence of the Rees-Stanojević sum under the condition that the Fourier coefficients satisfy a certain limit.
2. This generalizes a previous result by Garrett and Stanojević which proved convergence in the L-metric for quasi-convex sequences.
3. As a corollary, the author shows that the L1-convergence of the partial sums of the cosine series is equivalent to the condition that the product of the largest neglected coefficient and log n approaches 0 as n approaches
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document discusses the visual elements of images, identifying dots, lines, planes, color, and texture as the basic building blocks. It provides details on each element, including that a dot is the smallest and simplest, a line is a dot in motion, and planes are larger flat shapes. These elements can be used alone or combined to create images and convey ideas through visual expression.
The document analyzes and summarizes several movie posters. Key conventions noted across posters include using characters' faces showing different expressions to convey emotion, prominent use of red and brown colors, inclusion of a tagline, and inclusion of subtle hidden details within images. The document also provides analyses of individual posters, noting design elements like contrasting colors, symbolic imagery, and how aspects are intended to intrigue viewers without revealing full plot details.
This document is an invitation for bids for a contract related to the Electric Power Reconstruction II Project in Bosnia and Herzegovina. The contract is for Technical Monitoring of the Kukovi landslide area over a 2 year period. The scope of work includes setting up geodetic points and piezometers, conducting geodetic monitoring, water sampling, and producing technical reports. Interested and eligible bidders can purchase the full bidding documents for 100 EUR and submit bids by May 20, 2010. The criteria for bid evaluation will be bid price and criteria for responsiveness include the time schedule and meeting technical specifications.
The document discusses the basic visual elements of images: dots, lines, planes, color, and texture. It defines a dot as the smallest visual element, and a line as a dot in motion. Lines can be simple, composed of one stroke, or complex, composed of multiple line fragments. Together, these basic elements form a visual language that artists use to convey ideas, sensations and feelings.
El documento habla sobre la importancia de los rincones en el aula de educación infantil. Los rincones permiten a los niños explorar y aprender a través del juego en áreas como la lectura, el arte, la ciencia y la dramatización. Estos espacios flexibles ayudan a los niños a desarrollar su creatividad, autonomía e intereses individuales.
El documento habla sobre la importancia de los rincones en el aula de educación infantil. Los rincones permiten a los niños explorar y aprender a través del juego en áreas como la lectura, el arte, la ciencia y la dramatización. Estos espacios flexibles ayudan a los niños a desarrollar su creatividad, autonomía e intereses individuales.
El documento habla sobre la importancia de los rincones en el aula de educación infantil. Los rincones permiten a los niños explorar y aprender a través del juego en áreas como la literatura, el arte, la naturaleza y las matemáticas. Los rincones deben estar bien organizados y rotular para que los niños sepan qué actividades pueden realizar en cada uno.
The document discusses the production of a film promotional campaign that included a website, poster, and trailer. It describes how each media text was created, the software used, and evaluations of the campaign from test viewers. The website was designed to seem like an authentic town website to promote the film's setting and story. Feedback was generally positive but provided suggestions for improvements to the quality of images, sound, and navigation across the different promotional materials.
This document provides links to 4 videos and a website about Ellis Island. The first video is a 4 minute video about Ellis Island. The second link is an 8.5 minute video of a visit to Ellis Island. The third link is a 4 minute commentary video about Ellis Island. The final link is to a website where users can research immigrants who passed through Ellis Island.
This production schedule outlines work to be done over multiple days in February, September and October 2009, with tasks scheduled on the 9th, 22nd and 30th of September and the 1st and 9th of October.
The document describes 20 shots for a film involving a protagonist investigating a mysterious symbol. Shot 1 introduces the town of Judicia. Shots 2-3 show a news report of a death. Shots 4-6 depict the protagonist watching TV and someone driving towards a man. Shots 7-9 show an axe attack, a gun, and a crowd around a corpse. Shots 10-12 show the protagonist noticing a mark on the corpse and analyzing it. Shots 13-14 show research on a laptop and website. Shots 15-16 feature the symbol on a mirror startling the protagonist. Shots 17-19 involve smashing a bottle, dodging a bullet, and fighting an antagonist. The final shot shows
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DOI: 10.13140/RG.2.2.24591.92329/9
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1. Repunit Lehmer numbers
Javier Cilleruelo
Instituto de Ciencias Matem´ticas (CSIC-UAM-UC3M-UCM)
a
and Departamento de Matem´ticas, Facultad de Ciencias
a
Universidad Aut´noma de Madrid
o
28049, Madrid, Espa˜a n
franciscojavier.cilleruelo@uam.es
Florian Luca
Instituto de Matem´ticas
a
Universidad Nacional Autonoma de M´xico
e
C.P. 58089, Morelia, Michoac´n, M´xico
a e
fluca@matmor.unam.mx
Abstract
A Lehmer number is a composite positive integer n such that φ(n) |
n − 1. In this paper, we show that given a positive integer g > 1 there
are at most finitely many Lehmer numbers which are repunits in base
g, and they are all effectively computable. Our method is effective and
we illustrate it by showing that there is no such Lehmer number when
g ∈ [2, 1000].
2000 Mathematics Subject Classification: Primary 11N25, Secondary 11B39
Keywords: Lehmer numbers, repunits
1 Introduction
Let φ(n) be the Euler function of the positive integer n. Clearly, φ(n) = n−1
if n is a prime. Lehmer [4] (see also B37 in [3]) conjectured that if φ(n) | n−1,
then n is prime. To this day, no counterexample to this conjecture has been
found. A composite number m such that φ(m) | m − 1 is called a Lehmer
number. Thus, Lehmer’s conjecture is that Lehmer numbers don’t exist but
it is not even known that there should be at most finitely many of them.
Given a positive integer g > 1 a base g repunit is a number of the form
m = (g n − 1)/(g − 1) for some integer n ≥ 1. We will refer to such numbers
1
2. simply as repunits without mentioning the dependence on g. It is not known
whether given g there are infinitely many repunit primes. When g = 2 such
primes are better known as Mersenne primes. In [5], it was shown that there
is no Lehmer number in the Fibonacci sequence. Here, we use some ideas
from [5] together with finer arguments to prove the following results. In
what follows, we write un = (g n − 1)/(g − 1).
Theorem 1. For each fixed g > 1, there are only finitely many positive inte-
gers n such that un is a Lehmer number, and all are effectively computable.
Theorem 2. There is no Lehmer number of the form un when 2 ≤ g ≤ 1000.
Acknowledgement. We thank the anonymous referee for numerous comments
which improved the quality of this paper. Work on this paper was done during a
pleasant visit of F. L. at the Mathematics Department of the Universidad Aut´noma
o
de Madrid in Spring of 2008. He thanks the people of that Institution for their
hospitality. J. C. was supported in part by Project MTM2005-04730 from MEC
(Spain) and the joint Madrid Region-UAM project TENU2 (CCG07-UAM/ESP-
1814). F. L. was also supported in part by Grants SEP-CONACyT 79685 and
PAPIIT 100508.
2 Preliminaries
For a prime q and a nonzero integer m we write νq (m) for the exponent
of q in the factorization of m. We start by collecting some elementary and
well-known properties of the sequence of general term un = (g n − 1)/(g − 1)
for n ≥ 1.
Lemma 1. i) un = g n−1 + · · · + g + 1. In particular, un is coprime to g.
ii) The sequence un satisfies the linear recurrence
u1 = 1, un = gun−1 + 1, n ≥ 2. (1)
iii) If d | n, then ud | un .
iv) Let q be a prime. If q | n, then q | φ(un ).
v) Let q be a prime not dividing g. If q | n, then νq (un−1 ) ≤ νq (uf ) ≤
νq (uq−1 ), where f is the order of g modulo q.
vi) If un is a Lehmer number, then (un , g − 1) = 1.
2
3. Proof. i) and ii) are obvious. For iii), we observe that
gn − 1 (g d )n/d − 1 g d − 1 n
un = = · = (g d ) d −1 + · · · + 1 ud .
g−1 gd − 1 g−1
iv) If q = 2, then un ≥ u2 = g + 1 > 2, therefore φ(un ) is even. Assume
now that q is odd. Let p be a prime which divides uq . Then, g q ≡ 1 (mod p),
so the order of g modulo p is 1 or q. If it is q, then q | p − 1 | φ(uq ). Since
by iii ) we know that uq | un , we get that q | φ(uq ) | φ(un ), which is what
we wanted. Assume now that the order of g modulo p is 1 for all primes p
dividing uq . Let us show that this cannot happen. If it would, then p | g − 1
for all such primes p. Since also p | uq , we have
gq − 1
0 ≡ uq ≡ = g q−1 + · · · + g + 1 ≡ 1 + · · · + 1 + 1 ≡ q,
g−1
where all congruences above are modulo p. Thus, p | q, therefore p = q.
Hence, uq = q α for some positive integer α. However, writing g − 1 = qλ
with some positive integer λ, we get
uq = (1 + qλ)q−1 + (1 + qλ)q−2 + · · · + (1 + qλ) + 1
≡ (1 + (q − 1)qλ) + (1 + (q − 2)qλ) + · · · + (1 + qλ) + 1 (mod q 2 )
≡ q + qλ((q − 1) + · · · + 1) (mod q 2 )
q 2 (q − 1)λ
≡ q+ (mod q 2 )
2
≡ q (mod q 2 ).
In the above chain of congruences, we used the fact that q is odd, therefore
(q − 1)/2 is an integer. The above argument shows that q uq ; hence, α =
1. So, uq = q. However, we clearly have uq ≥ 2q − 1 > q, which is a
contradiction.
v) We may also assume that q | un−1 , otherwise νq (un−1 ) = 0 and the
first inequality is clear. Now g n−1 ≡ 1 (mod q), and so f | n − 1. We now
write n−1
−1
un−1 = (g f ) f + · · · + 1 uf .
The quantity in brackets above is not divisible by q since it is congruent to
(n − 1)/f modulo q and q | n. Thus, νq (un−1 ) ≤ νq (uf ) ≤ νq (uq−1 ), where
the last inequality follows because f | q − 1, so, uf | uq−1 by iii).
vi) Suppose that q is a prime dividing both un and g − 1. We then have
that g ≡ 1 (mod q) and un = g n−1 + · · · + 1 ≡ n (mod q). Thus, q | n. By
3
4. iv), we know that q | φ(un ). Since un is a Lehmer number, we know that
φ(un ) | un − 1 = gun−1 . Since q divides g − 1, it cannot divide g, therefore
q | un−1 . Hence, q | un − un−1 = g n−1 , which is not possible.
In the next lemma, we gather some known facts about Lehmer numbers.
Lemma 2. i) Any Lehmer number must be odd and square-free.
K
ii) If m = p1 · · · pK is a Lehmer number, then K 2 > m.
iii) If m = p1 · · · pK is a Lehmer number, then K ≥ 14.
Proof. i) If m > 2 then φ(m) is even, and since φ(m) | m − 1, we get that
m must be odd. If p2 | m, then p | φ(m), and since φ(m) | m − 1, we have
p | m − 1, which is not possible. Part ii) was proved by Pomerance in [6],
while part iii) was proved by Cohen and Hagis in [2].
Lemma 3. Theorems 1 and 2 hold when g is even.
Proof. Note that
2K | (p1 − 1) · · · (pK − 1) = φ(un ) | un − 1 = gun−1 .
We observe that if g is even, then un−1 is odd. In that case, we have
K ≤ ν2 (φ(un )) ≤ ν2 (gun−1 ) = ν2 (g), (2)
implying, by Lemma 2 ii), that
K ν2 (g)
g n−1 < un < K 2 ≤ (ν2 (g))2 ≤ (ν2 (g))g .
Thus,
g log(ν2 (g))
n≤1+ .
log g
For Theorem 2, we observe that ν2 (g) ≤ 9 for any g ≤ 1000, and we obtain
a contradiction from (2) and Lemma 2 iii).
From Lemma 1 i), we see that if g is odd and n is even, then un is
even, so Lemma 2 i) shows that un cannot be a Lehmer number. From now
on, we shall assume that both g and n are odd and larger than 1 and that
un = (g n − 1)/(g − 1) is a Lehmer number. We also keep the notation:
α α
n = q1 1 · · · qs s , where 2 < q1 < · · · < qs (3)
are primes and α1 , . . . , αs are positive integers, and
un = p1 · · · pK , where 2 < p1 < · · · < pK (4)
are also primes.
4
5. 3 Proof of Theorem 1
3.1 Primitive divisors
Let (An )n≥1 denote a sequence with integer terms. We say that a prime p is
a primitive divisor of An if p | An and gcd(p, Am ) = 1 for all non-zero terms
Am with 1 ≤ m < n.
In 1886, Bang [1] showed that if g > 1 is any fixed integer, then the
sequence (An )n≥1 of nth term An = g n − 1 has a primitive divisor for any
index n > 6.
We will apply this important theorem to our sequence un .
Lemma 4. If d > 1 is odd, then ud has a primitive divisor pd . Furthermore,
pd ≡ 1 (mod 2d).
Proof. We revisit the argument already used at Lemma 1 iv). We write
vn = g n − 1. It is well-known that gcd(vn , vm ) = vgcd(n,m) . Observe also
that
vd
= ud = g d−1 + · · · + 1 ≡ d (mod g − 1),
v1
therefore if d is a prime not dividing g − 1, then vd has primitive divisors. If
d > 2 is a prime dividing g − 1, then the above argument, or the argument
from the proof of Lemma 1 iv ), shows that gcd(vd , v1 ) is a power of d. Write
g − 1 = dλ and observe that
vd
= (1 + dλ)d−1 + (1 + dλ)d−2 + · · · + 1
v1
≡ (1 + (d − 1)dλ) + (1 + (d − 2)dλ) + · · · + 1
= d + dλ((d − 1) + (d − 2) + · · · + 1) (mod d2 )
d2 (d − 1)
≡ d+ λ (mod d2 ) ≡ d (mod d2 ).
2
Thus, d vd /v1 , and therefore
vd 1
= (g d−1 + · · · + 1) > 1
dv1 d
is an integer coprime to v1 , so vd again has primitive divisors. Thus, v3 and
v5 (and, of course, v1 if g > 2) have primitive divisors. The fact that vd has
primitive divisors for all odd d ≥ 7 follows from Bang’s result.
We now note that if p is a primitive prime divisor of vd for d > 1, then
g d ≡ 1 (mod p), and d is the order of g (mod p). Indeed, for if not, then
5
6. f < d and p | vf , contradicting the fact that p is primitive for vd . So, d | p−1,
and since d is odd, we get that d | (p − 1)/2. Thus, p ≡ 1 (mod 2d).
Since a prime factor of g − 1 cannot be a primitive divisor for vd except
for d = 1, we deduce that if d > 1, then the primitive prime divisors for vd
are exactly those of ud = vd /(g − 1), and we get the first assertion of the
lemma.
In what follows, for a positive integer m we use ω(m) and τ (m) for the
number of prime divisors and the total number of divisors of m, respectively.
Lemma 5. If un is square-free, n is odd and (un , g − 1) = 1, then
un ω(n) q log g
log < 1 + log
φ(un ) 2q log(2q + 1)
τ (n) − 2 q 2 log g
+ 1 + log ,
2q 2 log(2q 2 + 1)
where q is the smallest prime dividing n.
Proof. We write Pd = {p is primitive prime divisor for ud }. We shall first
prove that
:= p = un .
1<d|n p∈Pd
To see the above formula, we observe that if p | ud and p g − 1, then p ∈ Pd
for some 1 < d | n. Since un is square-free, we have that un | . On the
other hand, the sets Pd are disjoint, and if p ∈ Pd , then p | ud | un . Thus,
| un .
Now, since un is square-free,
φ(un ) = (p − 1),
1<d|n p∈Pd
and then
un 1
log < .
φ(un ) p−1
d|n p∈Pd
d>1
Since all the primes p ∈ Pd are congruent to 1 (mod 2d), we have
#Pd
1 1 1 1
Sd := ≤ ≤ (1 + log #Pd ).
p−1 2d j 2d
p∈Pd j=1
6
7. To bound the cardinality of Pd , we observe that (2d + 1)#Pd ≤ ud < g d , so
d log g
#Pd < .
log(2d + 1)
We observe that d ≥ q and if d is not a prime, then d ≥ q 2 . Then
1 q log g
Sd = Sd + Sd ≤ ω(n) 1 + log
2q log(2q + 1)
1<d|n d|n d|n
d prime d composite
1 q 2 log g
+(τ (n) − 2) 1 + log .
2q 2 log(2q 2 + 1)
3.2 Bounds for q1 and τ (n)
Recall that we keep the notations from (3) and (4).
Lemma 6. If un is a Lehmer number and n is odd, then
αi (αi + 1) α
τ (n/qi ) ≤ τ (n/qi i ) ≤ νqi (φ(un )) ≤ νqi (gun−1 )
2
νqi (g), if qi |g;
≤ (5)
νqi (uqi −1 ), if qi g
for all i = 1, . . . , s.
Proof. Lemma 4 implies that for each divisor of n of the form qi d withα
αi
1 ≤ α ≤ αi and d | (n/qi ), the divisor uqi d of un has a primitive prime
α
factor pqi d ≡ 1 (mod dqi
α
α ). In particular, q α | p α − 1, and the primes p α
i dqi dqi
α
are distinct as d ranges over the divisors of n/qi i . Thus,
α
(1+···+αi )τ (n/qi i )
qi | (pdqi − 1) |
α (p − 1)
1≤α≤αi d|n/q αi p|un
i
= φ(un ) | un − 1 | gun−1 ,
which gives the two central inequalities. The first inequality is trivial and
the equality holds when αi = 1. When qi | g, the last inequality follows from
Lemma 1 i), while when qi g, then νqi (gun−1 ) = νqi (un−1 ), and we apply
Lemma 1 v) to get the desired conclusion.
7
8. √
Lemma 7. Let un be a Lehmer number with both n and g odd. If qi > g,
then
τ (n/qi ) ≤ qi − 2.
√
Proof. If qi | g and qi > g, then νqi (g) = 1, and Lemma 6 above gives
τ (n/qi ) ≤ νqi (g) = 1 ≤ qi − 2. (6)
If qi g, then, again by Lemma 6 above, we have
τ (n/qi ) ≤ νqi (uqi −1 ).
Observe that
uqi −1 | g qi −1 − 1 = g (qi −1)/2 − 1 g (qi −1)/2 + 1 .
Since qi cannot divide both factors above, we have that
τ (n/qi ) ≤ νqi (g (qi −1)/2 + ) for some ∈ {−1, +1}.
If τ (n/qi ) ≥ qi − 1, then
qi i −1 ≤ qi
q τ (n/qi )
≤ g (qi −1)/2 + 1 ≤ (qi − 1)(qi −1)/2 + 1,
2
(7)
and we get a contradiction for qi > 3, because
qi i −1 = ((qi − 1) + 1)(qi −1)/2
q 2
and the expression on the right is larger that (qi − 1)(qi −1)/2 + 1 except when
2
qi = 3.
2
Finally, if qi = 3, the only odd g < qi with qi g are g = 5 and g = 7.
But in both cases we have τ (n/3) ≤ ν3 (u2 ) ≤ 1 ≤ qi − 2, which completes
the proof of this lemma.
Lemma 8. Let un be a Lehmer number with both n and g odd. Then
√
q1 ≤ max{ g, 19}. (8)
Proof. Assume that the above inequality does not hold. Then q1 ≥ 23,
2 √
g ≤ q1 − 1, and since q1 > g, we can apply Lemma 7 to deduce that
τ (n) ≤ 2τ (n/q1 ) ≤ 2q1 − 4. We also observe that τ (n) ≥ 2ω(n) , so ω(n) ≤
log(2q1 − 4)/ log 2.
8
9. Since un is a Lehmer number, we have that 2 ≤ un /φ(un ). Now Lemma
5 and the bounds above give
log ((2q1 − 4)/ log 2) 2
q1 log(q1 − 1)
log 2 < 1 + log
2q1 log(2q1 + 1)
2q1 − 6 q 2 log(q 2 − 1)
+ 2 1 + log 1 2
1
,
2q1 log(2q1 + 1)
which is false for q1 ≥ 23.
For a given value of g, Lemma 8 gives us our bound for q1 and then
this is used in Lemma 6, since τ (n) ≤ 2τ (n/q1 ), to give a bound for τ (n).
Observe also that ω(n) ≤ log τ (n)/ log 2.
3.3 The conclusion of the proof of Theorem 1
Since we have already proved that both s = ω(n) and τ (n) are bounded by
effectively computable constants depending only on g, in order to conclude
the proof of Theorem 1 it is enough to prove that all the primes qi with
i = 1, . . . , s are also bounded by effectively computable constants depending
on g. We shall prove this by induction on i = 1, . . . , s observing that this has
already been achieved for i = 1. Let i ≤ s − 1 and assume that qi has been
j=i α
bounded. Put Qi = j=1 qj j . There are only finitely many possibilities for
this number. We put gi = g Qi , ni = n/Qi and rewrite the condition that un
is Lehmer as
n n
g Qi − 1 gi i − 1 g Qi − 1 gi i − 1
aφ · = un − 1 = · −1
g−1 gi − 1 g−1 gi − 1
m
with some integer a ≥ 2. We put wm = (gi − 1)/(gi − 1) for the sequence
of repunits in base gi . Then, since un is square-free, we get that
aφ(uQi )φ(wni ) = uQi wni − 1,
therefore
φ(uQi ) wni 1
a = − . (9)
uQi φ(wni ) uQi φ(wni )
The left hand side takes only finitely many values, which are all effectively
computable. Assume that it takes some value δ ≤ 1. Then
1
wni − 1 < wni − = δφ(wni ) ≤ φ(wni ),
uQi
9
10. a contradiction. Thus, it remains to study the case when the right hand side
in (9) is > 1. Let δi > 1 be the smallest possible value larger than 1 of the
left hand side of (9). Clearly, this is effectively computable. We then get
wni
δi < .
φ(wni )
We observe that wni is a sequence “like” un but the new value of g is
gi = g Qi and the new value of n is ni = n/Qi . Thus, the smallest prime
factor of ni is qi+1 . We also note that τ (ni ) = τ (n/Qi ) < τ (n) which is
bounded, and that ω(ni ) < ω(n). Finally, we observe that (wni , g Qi −1) = 1,
otherwise, since (wni , g−1) = 1, the number un = (g Qi −1)wni /(g−1) would
not be square-free.
We now apply Lemma 5 to obtain that
ω(ni ) Qi qi+1 log g
log δi < 1 + log
2qi+1 log(2qi+1 + 1)
2
Qi qi+1 log g
τ (ni ) − 2
+ 2 1 + log 2 . (10)
2qi+1 log(2qi+1 + 1)
log qi+1
Hence, log δi qi+1 , where the constant implied by the Vinogradov sym-
bol above depends only on g, implying that qi+1 must be bounded by
some effectively computable constant depending only on g. This concludes
the proof of Theorem 1.
4 Proof of Theorem 2
We assume that g is odd and 3 ≤ g ≤ 999, so that 3 ≤ q1 ≤ 31 by Lemma
8.
Claim 1: That νq1 (uq1 −1 ) ≤ 5 can be checked with Mathematica. In
particular, by Lemma 6, we have that if q1 g, then νq1 (φ(un )) ≤ 5.
Claim 2: τ (n/q1 ) ≤ νq1 (φ(un )) ≤ 6, and s ≤ 3.
Suppose first that q1 | g. Then, by Lemma 6,
log g log 1000
τ (n/q1 ) ≤ νq1 (φ(un )) ≤ νq1 (gun−1 ) = νq1 (g) ≤ ≤ = 6.
log q1 log 3
In the above, in fact νq1 (g) < 6 unless (q1 , g) = (3, 729). Then, for any q1 ,
by Claim 1, either q1 = 3 and τ (n/q1 ) ≤ 6, or τ (n/q1 ) ≤ 5. In particular,
τ (n) ≤ 2τ (n/q1 ) ≤ 12, which shows that s ≤ 3.
10
11. Claim 3: s ≥ 2.
Let us see indeed that for our particular case we cannot have s = 1. If
α
this were so, then n = q1 1 . Then each prime factor pj of un is primitive
for some divisor d > 1 of n, which is a power of q1 (again, this is because
gcd(un , g − 1) = 1). Thus, pj ≡ 1 (mod q1 ) for all j = 1, . . . , K, showing
that νq1 (φ(un )) ≥ K ≥ 14 (see Lemma 2 iii)), which contradicts the fact
that νq1 (φ(un )) ≤ 6. Hence, s ≥ 2.
Claim 4: α1 = 1 except when (α1 , q1 , g) = (2, 3, 729).
α
Put again, as in the proof of Theorem 1, Q1 = q1 1 . By Lemma 6 and
the fact that s ≥ 2, we have
α1 (α1 + 1) α
α1 (α1 + 1) ≤ τ (n/q1 1 ) ≤ νq1 (φ(un )).
2
By Claims 1 and 2 above, we know that νq1 (φ(un )) ≤ 5, except when
(α1 , q1 , g) = (2, 3, 729). So, α1 = 1 except for this case.
Note that, at any rate, since s ≥ 2, it follows that 2 ≤ τ (n/q1 ) ≤
νq1 (guq1 −1 ). A computation with Mathematica revealed 431 possibilities for
the pairs (q1 , g) in our range satisfying νq1 (guq1 −1 ) ≥ 2.
Claim 5: q2 ≤ 19.
The smallest left hand side in (9) computed over all the 432 possible
pairs (Q1 , g) has δ1 > 1.49 (it was obtained for g = 809, Q1 = q1 = 3 and
a = 2, for which the obtained value is > 1.495). Of course, we did not
factor all the numbers of the form (g Q1 − 1)/(g − 1). If q1 = 31, then the
smallest prime p1 ≡ 1 (mod q1 ) is 311. The number K of prime factors of
u31 satisfies therefore
log uq1 3 · 31 · log 10
K< < < 38;
log p1 log 311
hence,
37
φ(uq1 ) 1
a ≥2 1− > 1.7.
uq1 311
Similarly, using the fact that when q1 = 29 and 23 the first two primes
congruent to 1 (mod q1 ) are 59 and 233, and 47 and 139 respectively, and
3 · 29 · log 10 3 · 23 · log 10
< 37 and < 33,
log 233 log 139
11
12. we have that
36 32
φ(uq1 ) 1 1 1 1
a ≥ 2 min 1− 1− , 1− 1−
uq1 59 233 47 139
> 1.55,
whenever q1 ∈ {23, 29}. Thus, we have factored only the numbers uQ1 with
Q1 ≤ 19. We now use inequality (10) for i = 1 to obtain
ω(n1 ) Q1 q2 log g
log(1.49) < 1 + log
2q2 log(2q2 + 1)
τ (n1 ) − 2 2
Q1 q2 log g
+ 2 1 + log 2 .
2q2 log(2q2 + 1)
2
If q1 > 3, then Q1 = q1 ≤ 31. If q1 = 3, then Q1 = q1 = 9. Thus, Q1 ≤ 31
in both cases. We also saw in Claims 1 and 2 that τ (n1 ) ≤ τ (n/q1 ) ≤ 6, so
also ω(n1 ) ≤ 2. Hence,
1 31q2 log 999 2 2
31q2 log 999
log(1.49) < 1 + log + 2 1 + log 2 ,
q2 log(2q2 + 1) q2 log(2q2 + 1)
and this inequality does not hold when q2 ≥ 23.
4.1 The conclusion of the proof of Theorem 2
Thus, 3 ≤ q1 < q2 ≤ 19. The argument showing that α1 = 2 except if
(q1 , g) = (3, 729) now shows that α2 = 1. We are now able to show that
s = 2. Indeed, if it were not so, then we would have both τ (n/q1 ) ≥ 4
and τ (n/q2 ) ≥ 4. A quick computation with Mathematica shows that while
there are pairs (q, g) such that νq (guq−1 ) ≥ 4 in our ranges, there is no odd
g in [3, 999] that has the above property with respect to two different primes
3 ≤ q1 < q2 ≤ 19. Thus, either n = q1 q2 , or n = 9q2 and g = 729. To test
these last pairs, we proceeded as follows. First we have detected all pairs
(n, g) with n = q1 q2 with 3 ≤ q1 < q2 ≤ 19 and odd g ∈ [3, 999] such that
νqi (gun−1 ) ≥ 2 holds for both i = 1, 2. There are 2043 such pairs. For each
one of these we checked that ν2 (un−1 ) < 14. Similarly, when Q1 = 9 and
g = 729, the only possibility for q2 in our range such that νq2 (uq2 −1 ) ≥ 2 is
q2 = 11, but in this case n = 99 and ν2 (un−1 ) = 1 < 14. This finishes the
proof of Theorem 2.
12
13. References
[1] A. S. Bang, “Taltheoretiske Undersølgelser”, Tidskrift f. Math. 5
(1886), 70–80 and 130–137.
[2] G. L. Cohen and P. Hagis, “On the number of prime factors of n if
φ(n) | n − 1”, Nieuw Arch. Wisk. (3) 28 (1980), 177–185.
[3] R. K. Guy, Unsolved Problems in Number Theory, Springer, 2004.
[4] D. H. Lehmer, “On Euler’s totient function”, Bull. Amer. Math. Soc.
38 (1932), 745–751.
[5] F. Luca, “Fibonacci numbers with the Lehmer property”, Bull. Polish
Acad. Sci. Math. 55 (2007), 7–15.
[6] C. Pomerance, “On composite n for which φ(n) | n − 1, II”, Pacific J.
Math. 69 (1977), 177–186.
13