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