The Thue-Siegel-Roth Theorem (original) (raw)

2008

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Abstract

In this paper we will give a proof of the Thue-Siegel-Roth Theorem, which states that for any algebraic number α and any ǫ > 0 there exists only a finite number of pairs of coprime integers p, q such that α − p q < 1 q 2+ǫ. We will follow the proof as it is presented Leveque's book, [8, ch 4]. This proof also deals with the more general case when p q is allowed to be an algebraic number in some fixed number field.

A STUDY ON ALGEBRAIC NUMBER THEORY AND ITS APPLICATIONS

Dr. M. Shanmuga Sundari* Dr. S. Sagathevan**

Current study explains the concept of Algebraic Number Theory and its applications. Study was based on the literature and descriptive in nature. Algebraic number theory is a rich and diverse subfield of abstract algebra and number theory, applying the concepts of number fields and algebraic numbers to number theory to improve upon applications such as prime factorization and primarily testing. In this study, researchers begin with an overview of algebraic number fields and algebraic numbers and then move into some important results of algebraic number theory, focusing on the quadratic, or Gauss reciprocity law.

Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

Journal of Number Theory, 1997

In a recent paper [2], M. Buck and D. Robbins have given the continued fraction expansion of an algebraic power series when the base field is F 3. We study its rational approximation property in relation with Roth's theorem, and we show that this element has an analog for each power of an odd prime number. At last we give the explicit continued fraction expansion of another classical example. §1. Introduction. Let K be a field. We denote K((T −1)) the set of formal Laurent series with coefficients in K. If α = k≤k 0 a k T k is an element of K((T −1)), with a k 0 = 0, we introduce the absolute value |α| = |T | k 0 and |0| = 0, with |T | > 1. It is well known that Roth's theorem (if α is an element of K((T −1)), irrational algebraic over K(T), then for all real ǫ > 0 we have |α − P/Q| > |Q| −(2+ǫ) for all P/Q) ∈ K(T) with |Q| large enough) fails if K has a positive characteristic p. In this case, which is the one we consider here, Liouville's theorem (there is a real positive constant C such that |α − P/Q| ≥ C|Q| −n for all P/Q ∈ K(T) , where n is the degree of α over K(T)) holds and is optimal. Many examples can be studied. A special case is the one where α satisfies an equation of the form α = (Aα p s + B)/(Cα p s + D) where A, B, C, D belong to K[T ], with AD−BC = 0, and s is a positive integer. Those elements have been studied by Baum and Sweet, Mills and Robbins, Voloch, de Mathan ([1],[5],[6],[7]). To simplify we will say that such an irrational algebraic element is an element of class I. It is also possible to study some particular rational functions, with coefficients in K[T ], of an element of class I (This was done by Voloch in [8]). For such simple examples, if d is a real number such that, for every ǫ > 0, we have |α − P/Q| > |Q| −(d+ǫ) for |Q| large enough, then there is a real positive constant C such that |α−P/Q| ≥ C|Q| −d , for all P/Q. But all these examples seem to be exceptions. It seems that , except for "particular" elements, Roth's theorem holds, and for an irrational algebraic element , for all ǫ > 0, we have |α − P/Q| > |Q| −(2+ǫ) , for |Q| large enough but not |α − P/Q| ≥ C|Q| −2 for all P/Q. Nevertheless, no algebraic element α, for which this result could be established, was known. It has only been proved that if α is an algebraic element of degree n, not of class I, then Thue's theorem holds, i.e. |α − P/Q| > |Q| −([n/2]+ǫ) , for |Q| large enough ([3]).

Primes in denominators of algebraic numbers

Cornell University - arXiv, 2022

Denote the set of algebraic numbers as Q and the set of algebraic integers as Z. For γ ∈ Q, consider its irreducible polynomial in Z[x], F γ (x) = a n x n + • • • + a 0. Denote e(γ) = gcd(a n , a n−1 ,. .. , a 1). Drungilas, Dubickas and Jankauskas show in a recent paper that Z[γ] ∩ Q = {α ∈ Q | {p | v p (α) < 0} ⊆ {p | p|e(γ)}}. Given a number field K and γ ∈ Q, we show that there is a subset X(K, γ) ⊆ Spec(O K), for which O K [γ] ∩ K = {α ∈ K | {p | v p (α) < 0} ⊆ X(K, γ)}. We prove that O K [γ] ∩ K is a principal ideal domain if and only if the primes in X(K, γ) generate the class group of O K. We show that given γ ∈ Q, we can find a finite set S ⊆ Z, such that for every number field K, we have X(K, γ) = {p ∈ Spec(O K) | p ∩ S = ∅}. We study how this set S relates to the ring Z[γ] and the ideal D γ = {a ∈ Z | aγ ∈ Z} of Z. We also show that γ 1 , γ 2 ∈ Q satisfy D γ1 = D γ2 if and only if X(K, γ 1) = X(K, γ 2) for all number fields K.

Elements of Analytic Number Theory

2013

Contents 6. Expression of the integral Chebyshev function via the Riemann function § 2.3. Riemann function: Analytic properties 1. Analytic extension of the Riemann function 2. Zeros of the Riemann function 3. Estimates of the logarithmic derivative 4. Proof of the Prime Number Theorem § 2.4. Problems Chapter 3. Dirichlet Theorem § 3.1. Finite abelian groups and groups of characters 1. Finite abelian groups 2. Characters 3. Characters modulo m § 3.2. Dirichlet series 1. Convergence of L-series 2. Landau Theorem 3. Proof of the Dirichlet Theorem Chapter 4. p-adic numbers § 4.1. Valuation fields 1. Basic properties 2. Valuations over rationals 3. The replenishment of a valuation field § 4.2. Construction and properties of p-adic fields 1. Ring of p-adic integers and its properties 2. The field of p-adic rationals is the replenishment of rationals in p-adic metric 3. Applications § 4.3. Problems Bibliography Glossary Index CHAPTER 1 Algebraic and transcendental numbers § 1.1. Field of algebraic numbers. Ring of algebraic integers 1. Preliminary information. Let us recall some basic notions from Abstract Algebra. Throughout we use the following notations:

On the Analytical Properties of Prime Numbers

IntechOpen's , 2023

In this work we have studied the prime numbers in the model P ¼ am þ 1, m, a>1∈ . and the number in the form q ¼ mam þ bm þ 1 in particular, we provided tests for hem. This is considered a generalization of the work José María Grau and Antonio M. Oller-marcén prove that if Cmð Þ¼ a mam þ 1 is a generalized Cullen number then ma m - ð Þ1 a ð Þ mod Cmð Þ a . In a second paper published in 2014, they also presented a test for Broth’s numbers in Form kpn þ 1 where k<p n . These results are basically a generalization of the work of W. Bosma and H.C Williams who studied the cases, especially when p ¼ 2, 3, as well as a generalization of the primitive MillerRabin test. In this study in particular, we presented a test for numbers in the form mam þ bm þ 1 in the form of a polynomial that highlights the properties of these numbers as well as a test for the Fermat and Mersinner numbers and p ¼ ab þ 1 a, b>1∈  and p ¼ qa þ 1 where q is prime odd are special cases of the number mam þ bm þ 1 when b takes a specific value. For example, we proved if p ¼ qa þ 1 where q is odd prime and a>1∈  where πj ¼ 1 q q j   then Pq2 j¼1 πjð Þ Cmð Þ a qj1 q  a m ð Þ - χð Þ m,qam ð Þ mod p Components of proof Binomial the- orem Fermat’s Litter Theorem Elementary algebra.

Algebraic Number Theory and Fermat's Last Theorem

The project aims to deliver sufficient mathematical background to understand a partial proof, due to Ernst Kummer, of Fermat's last theorem for a specific class of primes called regular primes. In doing so, we also develop some theory that is applicable to a wide range of scenarios in modern number theory, a few of which are discussed in somewhat more superficial detail in the final section.

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References (8)

  1. E. Landau, "Vorlesungen über Zahlentheorie Aus der elementaren Zahlentheorie", Chelsea Pub- lishing Company, 1946.
  2. G. H. Hardy and E. M. Wright, "An Introduction to the Theory of Numbers", Oxford University Press, 1979.
  3. K. Ireland and M. Rosen, "A Classical Introduction to Modern Number Theory", Springer, 1998.
  4. A. Thue, "J. Reine Angew. Math.", 135 (1909), 284-305.
  5. C. L. Siegel, Approximation algebraiseher Zahlen, Mathematische Zeitschrift, 10 (1921), 173-213.
  6. F. J. Dyson, The approximation to algebraic numbers by rationals, Acta Mathematica, 79 (1947), 225-240.
  7. K. F. Roth, Rational approximations to algebraic numbers, Mathematika, 2 (1955), 1-20.
  8. W. J. LeVeque, "Topics In Number Theory, Vol. I & II", Addison-Wesley Publishing Company, Inc, 1956.

Some theorems on diophantine approximation

Transactions of the American Mathematical Society, 1966

Introduction. The study of the values at rational points of transcendental functions defined by linear differential equations with coefficients in Q[z] (2) can be traced back to Hurwitz [1] who showed that if ,. , 1 z 1 z2 Az) = l+-b-lT+WTa)2l +where « is a positive integer, b is an integer, and b\a is not a negative integer, then for all nonzero z in Q((-1)1/2) the number y'(z)jy(z) is not in g((-1)1/2). Ratner [2] proved further results. Then Hurwitz [3] generalized his previous results to show that if nZ) ,+g(0) 1! +g(0)-g(l)2! + where f(z) and g(z) are in Q[z\, neither f(z) nor g(z) has a nonnegative integral zero, and degree (/(z)) < degree (g(z)) = r, then for all nonzero z in the imaginary quadratic field Q((-n)1'2) two of the numbers y(z),y(l)(z),-,yir\z) have a ratio which is not in Q((-n)112). Perron [4], Popken [5], C. L. Siegel [6], and K. Mahler [7] have obtained important results in this area. In this paper we shall use a generalization of the method which was developed by Mahler [7] to study the approximation of the logarithms of algebraic numbers by rational and algebraic numbers. Definition. Let K denote the field Q((-n)i/2) for some nonnegative integer «. Definition. For any monic 0(z) in AT[z] of degree k > 0 and such that 6(z) has no positive integral zeros we define the entire function oo d f(z)= £-. d^O d n oc«)

Some conjectures in elementary number theory

2013

We announce a number of conjectures associated with and arising from a study of primes and irrationals in mathbbR\mathbb{R}mathbbR. All are supported by numerical verification to the extent possible.

Some Results on Number Theory and Analysis

Mathematics and Statistics, 2022

In this work we obtain bounds for the sum of the integer solutions of quadratic polynomials of two variables of the form P = (10x + 9)(10y + 9) or P = (10x + 1)(10y + 1) or P = (10x + 7)(10y + 3) where P is a given natural number that ends in one. This allows us to decide the primality of a natural number P that ends in one. Also we get some results on twin prime numbers. In addition, we use special linear functionals defined on a real Hilbert space of dimension n, n ≥ 2 , in which the relation is obtained: a 1 + a 2 + • • • + a n = λ[a 2 1 + • • • + a 2 n ], where a i is a real number for i = 1, ..., n. When n = 3 or n = 2 we manage to address Fermat's Last Theorem and the equation x 4 + y 4 = z 4 , proving that both equations do not have positive integer solutions. For n = 2, the Cauchy-Schwartz Theorem and Young's inequality were proved in an original way.

Some conjectures in elementary number theory, II

2014

We announce a number of conjectures associated with and arising from a study of primes and irrationals in R. All are supported by numerical verification to the extent possible. This is an unpublished updated version as of August 13, 2014.

MY PROOFS OF CONJECTURES ON NUMBER THEORY -V ersion 1. (May 2025)

In this booklet, I present my proofs of open conjectures on the theory of numbers. It concerns the following conjectures: - The Riemann Hypothesis. - Beal's conjecture. - The conjecture c<rad1.63(abc)c<rad^{1.63}(abc)c<rad1.63(abc). - The explicit abcabcabc conjecture of Alan Baker. - Two proofs of the abcabcabc conjecture. - The conjecture c<rad2(abc)c<rad^2(abc)c<rad2(abc).

Simultaneous Diophantine Approximation with Primes

Journal of the London Mathematical Society, 1989

Many variations of the theorems of Dirichlet and Kronecker in Diophantine approximation [3, Chapters 11.3 and 23] have been established, by restricting the class of integers allowed in the problem. For example, see [1], where one is restricted to the values taken by a polynomial with integer coefficients. Another example is afforded by the following theorem. Suppose that a is irrational and let \\y\\ denote the smallest distance of y from an integer. Then, for any real number ft, there are infinitely many primes p such that \\ap-p\\<p-3 ' 10. (1.1) This result was established in [5], improving earlier results of Vinogradov [11, Chapter 11] and Vaughan [9]. In this paper we shall consider extending this result to simultaneous approximation. In [4, Chapter 3] the author considered this problem for sets of not very well approximable numbers (see [7], or below for the definition) and this paper develops the author's previously unpublished work. Balog and Friedlander have recently done some work on this question [2], but our results supersede theirs in all cases. We wish to prove that max is ' small' for infinitely many primes p. A necessary restriction on a 15 ..., OL S is given by the following definition due to Balog and Friedlander. A set of real numbers {a l5 ..., a s } is called compatible if s s £ «, a, e Q implies that ^^a^e Z whenever n lt ...n s are integers. Balog and Friedlander then proved the following. Let {a 15 ..., a s } be a compatible set of real algebraic numbers, lying in a field of degree d. Then, for any A < (3ds + d-s-\)~1, there are infinitely many solutions in primes p to max || pa, || <p~A. We improve this result as follows.

The Trace Problem for Totally Positive Algebraic Integers

Journal of the Australian Mathematical Society, 2011

Suppose that P (x) = x d + a 1 x d−1 + · · · + a d is a polynomial with integer coefficients, irreducible, all whose roots are real and positive. In a remarkable paper of 1918, I. Schur proved that if c < √ e, then there are only finitely many such polynomials for which the average of its roots, equal to −a 1 /d, is less than c. The Schur-Siegel-Smyth trace problem asks for the largest value of c for which the same conclusion holds. In this paper we give an account of the history of the problem, the latest results, and its relations with other problems in number theory.