Quadratic Algebraic Number Research Articles

Representations of a number in an arbitrary base with unbounded digits

Abstract In this paper, we prove that, for β ∈ ℂ {\beta\in{\mathbb{C}}} , every α ∈ ℂ {\alpha\in{\mathbb{C}}} has at most finitely many (possibly none at all) representations of the form α = d n ⁢ β n + d n - 1 ⁢ β n - 1 + … + d 0 {\alpha=d_{n}\beta^{n}+d_{n-1}\beta^{n-1}+\dots+d_{0}} with nonnegative integers n , d n , d n - 1 , … , d 0 {n,d_{n},d_{n-1},\dots,d_{0}} if and only if β is a transcendental number or an algebraic number which has a conjugate over ℚ {{\mathbb{Q}}} (possibly β itself) in the real interval ( 1 , ∞ ) {(1,\infty)} . The nontrivial part here is to show that for every algebraic number β lying with its all conjugates in ℂ ∖ ( 1 , ∞ ) {{\mathbb{C}}\setminus(1,\infty)} , there is α ∈ ℚ ⁢ ( β ) {\alpha\in{\mathbb{Q}}(\beta)} with infinitely many such representations. In a particular case, when β is a quadratic algebraic number, this was recently established by Kala and Zindulka.

Sums of binomial coefficients evaluated at [formula omitted], and applications

The additive monoid R+(x) is defined as the set of all nonnegative integer linear combinations of binomial polynomials (xn) for n∈Z+. This paper is concerned with the inquiry into the structure of R+(α) for complex numbers α. Particularly interesting is the case of algebraic α which are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep(St) (and also Rep(GLt)) for t∈C﹨Z+. We prove that this object is a ring if and only if α is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by α, i.e. all elements of OQ(α), are also contained in this ring. We also give two explicit representations of R+(α) for both algebraic integers and general algebraic numbers α. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R+(α) for both quadratic algebraic numbers and roots of unity.

On a problem of Pethő

In this paper we deal with a problem of Pethő related to existence of a quartic algebraic integer α for whichβ=4α4α4−1−αα−1 is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many α's such that the corresponding β is quadratic. Moreover, we present a description of quartic numbers α such that the corresponding β is a quadratic real number.

Mosaic Supercongruences of Ramanujan Type

In this article, we present analogues of supercongruences of Ramanujan type observed by L. Van Hamme and W. Zudilin. Our congruences are inspired by Ramanujan-type series that involve quadratic algebraic numbers.

On the limit points of $(a_n\xi )_{n=1}^{\infty }$ mod $1$ for slowly increasing integer sequences $(a_n)_{n=1}^{\infty }$

In this paper, we are interested in sequences of positive integers (a n ) ∞ n-1 such that the sequence of fractional parts {a u ξ} ∞ n=1 has only finitely many limit points for at least one real irrational number ζ. We prove that, for any sequence of positive numbers (g n ) ∞ n=1 satisfying g n ≥ 1 and lim n→∞ g n = ∞ and any real quadratic algebraic number a, there is an increasing sequence of positive integers (a n ) ∞ n=1 such that a n ≤ n gn for every n ∈ N and lim n→∞ {a n α} = 0. The above bound on an is best possible in the sense that the condition lim n→∞ g n = ∞ cannot be replaced by a weaker condition. More precisely, we show that if (a n ) ∞ n=1 is an increasing sequence of positive integers satisfying lim inf n→∞ a n /n < oo and ζ is a real irrational number, then the sequence of fractional parts {anξ} ∞ n=1 has infinitely many limit points.

Integral Points on Certain Elliptic Curves

By using algebraic number theory method and p-adic analysis method, we find all integral points on certain elliptic curves \_y\_2=(x+a)(\_x\_2+bx+c), a,b,\_c\_∈Z, b\_2<4\_c. Furthermore, we can find all integer solutions of certain hyperelliptic equations \_D\_\_y\_2=\_A\_\_x\_4+\_B\_x2+C, B\_2<4\_AC. As a particular example, we give a complete solution of the equation which was proposed by Zagier \_y\_2=x\_3-9\_x+28 by this method. In Appendix I and Appendix II, we give the computational method of finding the fundamental unit and factorizing quadratic algebraic number in the subring of a totally complex quartic field, respectively .

On the approximation of quadratic algebraic numbers

In the paper we construct such second order linear recursive sequences G and H of rational integers that with their terms |a -Gn+1 /H n| &lt; 1/ (\sqrtvDH2n) holds for every positive integer n, where a denotes a real quadratic algebraic integer of discriminant D. An approximating sequence of the form Gn+1 /Hn is also given for a if it is only a real quadratic algebraic number (not an algebraic integer), but in this case the approximating constant is not the best.

Quadratic algebraic numbers with finite b‐adic expansion on the unit circle and their distribution

AbstractWe consider elements x + y$ \sqrt {-m} $ in the imaginary quadratic number field ℚ($ \sqrt {-m} $) such that the norm x2 + my2 = 1 and both x and y have a finite b–adic expansion for an arbitrary but fixed integer base b. For m = 2, 3, 7 and 11 a full description of this set is given. Ordered by the number of digits in the b–adic expansion of the coordinates, the corresponding sequence of points on the unit circle, if infinite, is uniformly distributed. This continues work of P. Schatte who treated the case m = 1. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Weakly integrally closed domains:minimum polynomials of matrices

Must the coefficients of the minimum polynomial of a matrix over a domain lie in that domain? This question leads to the notion of a weakly integrally closed domain, over which the answer is ’yes’ for 3-by-3 matrices. It is shown that certain subalgebras of k[t] are weakly integrally closed, as are rings consisting of quadratic algebraic numbers.

Selberg trace formula for the Hilbert modular group of a real quadratic algebraic number field

In the paper the Selberg trace formula is derived for the Hilbert modular group of a real quadratic algebraic number field with one class of ideals, acting on the product of two upper semiplanes.