One More Construction Which Is Impossible

Every algebraic number except zero and unity (e), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in 7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

Every algebraic number except zero and unity (e°), when raised to a power which is an irrational algebraic number gives a transcendental number.This was conjectured by Hilbert, and proposed as a problem in 1900.fIt was only recently that Gelfond,{ and independently at about the same time, Schneider, § succeeded in proving this theorem.G. Ricci || has shown that the transcendentality persists if the algebraic numbers are replaced by products of algebraic numbers and a restricted type of Liouville number.The question we consider here is the more general one where the algebraic numbers are replaced by numbers capable of approximation by sequences of algebraic numbers in an appropriate manner.These form a restricted type of certain generalized Liouville transcendental numbers very closely related to those treated by Ore.^fThey bear a relation to the numbers of an algebraic field similar to that of the ordinary Liouville numbers to the rational numbers.Our principal result is that if X and Y are each suitably approximable by sequences of algebraic numbers, then XY and log X/log Y are transcendental.Some numerical examples are given in §7.Though the transcendental numbers found by us have the power of the continuum, they form a set of zero measure as we show.My interest in this subject was stimulated by a report on the Hilbert-Gelfond-Schneider theorem which I gave to the seminar of Professor H. Weyl at Princeton, and in particular by a conjecture of Professor J. von Neumann that natural generalizations of these results would involve approximations by algebraic numbers, rather than by rational numbers.I also wish to acknowledge my appreciation of the opportunity to work without distraction on this field, made possible by the courtesy of the Massachusetts Institute of Technology and the Institute for Advanced Study.

A set of complex numbers $S$ is called invariant if it is closed under addition and multiplication, namely, for any $x, y \in S$ we have $x+y \in S$ and $xy \in S$ . For each $s \in {\mathbb {C}}$ the smallest invariant set ${\mathbb {N}}[s]$ containing $s$ consists of all possible sums $\sum _{i \in I} a_i s^i$ , where $I$ runs over all finite nonempty subsets of the set of positive integers ${\mathbb {N}}$ and $a_i \in {\mathbb {N}}$ for each $i \in I$ . In this paper, we prove that for $s \in {\mathbb {C}}$ the set ${\mathbb {N}}[s]$ is everywhere dense in ${\mathbb {C}}$ if and only if $s \notin {\mathbb {R}}$ and $s$ is not a quadratic algebraic integer. More precisely, we show that if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is a transcendental number, then there is a positive integer $n$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for either $t=s$ or $t=s+s^2$ . Similarly, if $s \in {\mathbb {C}} \setminus {\mathbb {R}}$ is an algebraic number of degree $d \ne 2, 4$ , then there are positive integers $n, m$ such that the sumset ${\mathbb {N}} t^n+{\mathbb {N}} t^{2n} +{\mathbb {N}} t^{3n}$ is everywhere dense in ${\mathbb {C}}$ for $t=ms+s^2$ . For quadratic and some special quartic algebraic numbers $s$ it is shown that a similar sumset of three sets cannot be dense. In each of these two cases the density of ${\mathbb {N}}[s]$ in ${\mathbb {C}}$ is established by a different method: for those special quartic numbers, it is possible to take a sumset of four sets.

We begin with some preliminaries on algebraic numbers. Let α be an algebraic number. Then we observe that α satisfies the polynomial of minimal degree such that it has relatively prime integer coefficients and the leading coefficient positive. This is the minimal polynomial of α. The degree of this polynomial is called the degree of α. The maximum of the absolute values of the coefficients of this polynomial is called the height of α. We write ν = ν(α) for the least positive integer such that να is an algebraic integer i.e., να satisfies a monic polynomial with integer coefficients. The integer ν exists and we say that it is the denominator of α.

We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the non-denumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagonalization argument, which did not appear until 1891. The third proof is of the existence of real transcendental (i.e., non-algebraic) numbers. It also appeared in Cantor’s 1874 paper, as a corollary to the non-denumerability of the reals. What Cantor ingeniously showed is that the algebraic numbers are denumerable, so every open interval must contain at least one transcendental number.

A chromatic root is a root of the chromatic polynomial of a graph. Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008. The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

In this paper, we apply a combinatorial lemma to a well-known result concerning the transcendency of at least one of the numbers $\exp(\alpha_i\beta_j) (i=1, 2, 3; j=1, 2)$, where the complex numbers $\alpha_i,\beta_j$ satisfy linear independence conditions and show that for any $\alpha\neq0$ and any transcendental number $t$, we obtain that at most $\frac{1}{2}+(4N-4+\frac{1}{4})^{1/2}$ of the numbers $\exp(\alpha t^n)~(n=1,2,\ldots,N)$ are algebraic. Similar statements are given for values of the Weierstrass $\wp$-function and some connections to related results in the literature are discussed.

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures u ( s ) , u ( r ) u(s),u(r) with r r being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if s = r s = r or s = 1 − r s = 1 - r . We also show that for each positive integer n > 2 n{\text { > 2}} , there are algebraic integers and fractionals of degree n n in the unit interval that are binomial numbers.

We show that for all polynomials Q(x) with integer coefficients, that satisfy the extra condition |Q(0) · Q(1) | ≠ 1, there are infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). We also find some examples of polynomials Q(x) for which 1 is the only τ-number relative to the polynomial Q(x) and some examples of polynomials Q(x) with |Q(0) · Q(1)|= 1, which have infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). In addition, we prove one result about the generators of a τ-number.

In this paper, a method for image encryption is proposed based on a novel approach using chaos and transcendental numbers for generating the substitution box and the ciphering keys. The Chaotic Pi Ciphering (CPC) is a robust symmetric cryptosystem of the substitution-permutation-network type, that ciphers images in ten rounds. The objective of CPC is to avoid some general weakness points in cryptosystems, such as the vulnerability to certain types of attacks or the encryption with data loss. The number of rounds can be increased up to 14, with an even further increment of the ciphering security. An 8×8 substitution box is constructed using chaos, and its value of the Differential Power Analysis is lower than the corresponding to the Advanced Encryption Standard (AES). It makes CPC more resistant to differential and linear attacks than AES. CPC has approximately 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">512</sup> keys, presenting a high resistance to brute-force attacks. Chaos is produced by a nonlinear differential equation, and the Walsh function is applied to measure the nonlinearity of the box. The transcendental number pi and a positive integer are employed to generate the schedule keys, and the key length is equal to the image size. The randomness of the encrypted images is evaluated with four tests: entropy, correlation, Discrete Fourier Transform, and a goodness-of-fit test using the chi square distribution χ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . CPC was tested using well-known images from the related literature, and its performance was compared to the corresponding to AES. The results show a good entropy and a high randomness, indicating a high-quality ciphering.

Let u,v be solutions on an interval I of linear differential equations (LDEs) P=0 , Q=0 , respectively. We obtain a lower bound on the approximation of v by u in terms of bounds on the coefficients of LDE Si=0 (for several i ), satisfied by the i th derivative of v and by the i th derivative of a basis of the LDE P=0 . One could view this result as a differential analog of the Liouville theorem which states that two different algebraic numbers are well separated if they satisfy algebraic equations with small enough integer coefficients. Unlike the algebraic situation, in the differential setting, in order to bound from below the difference |u-v| , we need to involve not only the coefficients of P,Q themselves, but also those of Si .

For any irrational or transcendental number estimating of the quality of its approximation by rational fractions is one of the directions in the theory of Diophantine approximations. The quantitative characteristic of such approximation is called the measure (extent) of irrationality of the number. For almost a century and a half, scientists have developed various methods for evaluating the measure of irrationality and have obtained its values for a huge numberof irrational and transcendental numbers. Various approaches have been used to obtain the estimates and these approaches improved over time, leading to better estimates. The most commonly used method for obtaining such estimates is construction of linear forms with integer coefficients, which approximate a value, and studying of its asymptotic behavior. Approximating linear forms usually are constructed on the basis of continued fractions, Pad´e approximants,infinite series, and integrals. Methods for studying the asymptotics of such forms are currently quite standard, but the main problem is invention of a linear form with good approximating properties. The first estimates of the values of the arctangent function were obtained by M. Huttner in 1987 on the base of integral representation of the Gausss function. In 1993 A. Heimonen, T. Matala-Aho, K. Vaananen, using, like M. Huttner, Pad´e approximants for the Gaussianhypergeometric function, proved a general theorem for estimating of measures of irrationality of logarithms of rational numbers. Later, the same authors, using an approximating construction with Jacobi polynomials, obtained new estimates, in particular for the values of the function arctan 𝑥. Further research used various integral constructions, which allowed to get both general methods for arctan 𝑥 values and specialized methods for specific values. In the articles of E.B. Tomashevskaya, who in 2008 received a general estimate for the values of arctan 1/𝑛, 𝑛 ∈ N, was used a complex integral with the property of symmetry of integrand. This property played animportant role in obtaining the estimates, since it improved the asymptotic behavior of the coefficients of the linear form. Some integral constructions elaborated by other researchers alsohad different types of symmetry. In this article, we consider the main methods for estimating the values of the arctangent function, their features, research methods, and the best estimates at the moment.

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

This article reports on a study of high-school and of technologist students (prospective engineers and economists) understanding of real numbers. Our study was based on written response to a properly designed questionnaire and on interviews taken from students. The quantitative results of our experiment showed an almost complete failure of the technologist students to deal with processes connected to geometric constructions of incommensurable magnitudes. However, this didn’t prevent them in answering satisfactorily the other questions. The superiority of their correct answers with respect to those of high-school students, although the majority of them correspond to mediocre graduates of the secondary education, was evident in most cases. This is a strong indication that the age and the width of mathematical knowledge of the individual play an important role for the better understanding of the real numbers. The results of our experiment suggest also that the ability to transfer in comfort among several representations of real numbers helps students in obtaining a better understanding of them. A theoretical explanation about this is obtained through the adoption of the conceptual framework of dimensions of knowledge, introduced by Tirosh et al. (1998) for studying the comprehension of rational numbers. Following in part the idea of generic decomposition of the APOS analysis (Weller et al. 2009) we suggest a possible order for development of understanding the real numbers by students when teaching them at school. Some questions open to further research are also mentioned at the end of the paper. Keywords: Real, rational, irrational, algebraic and transcendental numbers, fractions, decimals, representations of real numbers.

Let t be any complex number different from the poles of a Weierstrass elliptic function ℘(z), having algebraic invariants. Then we estimate from below the sum where α and β are algebraic numbers. The estimate is given in terms of the heights of α and β and the degree of the field Q(α, β), where Q is the field of rationals.