scholarly journals Rational approximations to {\@root 3 \of {2}} and other algebraic numbers revisited

2007 ◽  
Vol 19 (1) ◽  
pp. 263-288 ◽  
Author(s):  
Paul Voutier
Keyword(s):  
Rational Approximations ◽  
Algebraic Numbers

Integer points on curves of genus 1

1970 ◽  
Vol 67 (3) ◽  
pp. 595-602 ◽  
Author(s):  
A. Baker ◽  
J. Coates
Keyword(s):  
Finite Number ◽  
Finite Basis ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Integer Points ◽  
Basis Theorem

1. Introduction. A well-known theorem of Siegel(5) states that there exist only a finite number of integer points on any curve of genus ≥ 1. Siegel's proof, published in 1929, depended, inter alia, on his earlier work concerning rational approximations to algebraic numbers and on Weil's recently established generalization of Mordell's finite basis theorem. Both of these possess a certain non-effective character and thus it is clear that Siegel's argument cannot provide an algorithm for determining all the integer points on the curve. The purpose of the present paper is to establish such an algorithm in the case of curves of genus 1.

scholarly journals New pancake series for π

2020 ◽  
Vol 104 (560) ◽  
pp. 296-303
Author(s):  
Yannick Saouter
Keyword(s):  
General Problem ◽  
Convergence Speed ◽  
Polynomial Form ◽  
Alternative Form ◽  
Rational Approximations ◽  
Algebraic Numbers

In [1], Dalzell proved that $\pi = \frac{{22}}{7} - \int_0^1 {\frac{{{t^4}{{(1 - t)}^4}}}{{1 + {t^2}}}}$ . He then used this equation to derive a new series converging to π. In [2], Backhouse studied the general case of integrals of the form $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}dt}$ and derived conditions on m and n so that they could be used to evaluate π. As a sequel, he derived accurate rational approximations of π. This work was extended in [3] where new rational approximations of π are obtained. Some related integrals of the forms $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{1 + {t^2}}}P(t)\,dt}$ and $\int_0^1 {\frac{{{t^m}{{(1 - t)}^m}}}{{\sqrt {1 - {t^2}} }}P(t)dt}$ with P(t) being of polynomial form are also investigated. In [4] the author gives more new approximations and new series for the case m = n = 4k. In [5] new series for π are obtained with the integral $\int_0^a {\frac{{{t^{12m}}{{(a - t)}^{12m}}}}{{1 + {t^2}}}dt}$ where $a = 2 - \sqrt 3$ . The general problem of improving the convergence speed of the arctan series by transformation of the argument has also been considered in [6, 7]. In the present work the author considers an alternative form for the denominators in integrals. As a result, new series are obtained for multiples of π by some algebraic numbers.

Simultaneous rational approximations to certain algebraic numbers

1967 ◽  
Vol 63 (3) ◽  
pp. 693-702 ◽  
Author(s):  
A. Baker
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Infinitely Many Solutions ◽  
Rational Approximations ◽  
Algebraic Numbers ◽  
Famous Theorem ◽  
Image Position

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the formis satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such thatThis result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

Rational approximations to algebraic numbers

Mathematika ◽  
1955 ◽  
Vol 2 (2) ◽  
pp. 160-167 ◽  
Author(s):  
H. Davenport ◽  
K. F. Roth
Keyword(s):  
Rational Approximations ◽  
Algebraic Numbers
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