Lower bounds for simultaneous Diophantine approximation constants

Werner Georg Nowak

Displaying similar documents to “Lower bounds for simultaneous Diophantine approximation constants”

On the critical determinants of certain star bodies

Werner Georg Nowak (2017)

Communications in Mathematics

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In a classic paper, W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body $| x_{1} | ({| x_{1} |}^{3} + {| x_{2} |}^{3} + {| x_{3} |}^{3}) \leq 1 .$ In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result...

A note on the number of $S$ -Diophantine quadruples

Florian Luca, Volker Ziegler (2014)

Communications in Mathematics

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Let $(a_{1}, \dots, a_{m})$ be an $m$ -tuple of positive, pairwise distinct integers. If for all $1 \leq i < j \leq m$ the prime divisors of $a_{i} a_{j} + 1$ come from the same fixed set $S$ , then we call the $m$ -tuple $S$ -Diophantine. In this note we estimate the number of $S$ -Diophantine quadruples in terms of $| S | = r$ .

Method of infinite ascent applied on $- (2^{p} \cdot A^{6}) + B^{3} = C^{2}$

Susil Kumar Jena (2013)

Communications in Mathematics

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In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for $(A, B, C)$ of the Diophantine equation $- (2^{p} \cdot A^{6}) + B^{3} = C^{2}$ for any positive integral values of $p$ when $p \equiv 1$ (mod 6) or $p \equiv 2$ (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.

On the diophantine equation $x^{2} + 5^{k} 17^{l} = y^{n}$

István Pink, Zsolt Rábai (2011)

Communications in Mathematics

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Consider the equation in the title in unknown integers $(x, y, k, l, n)$ with $x \geq 1$ , $y > 1$ , $n \geq 3$ , $k \geq 0$ , $l \geq 0$ and $gcd (x, y) = 1$ . Under the above conditions we give all solutions of the title equation (see Theorem 1).

Displaying similar documents to “Lower bounds for simultaneous Diophantine approximation constants”

On the critical determinants of certain star bodies

A note on the number of S -Diophantine quadruples

Method of infinite ascent applied on - ( 2 p · A 6 ) + B 3 = C 2

On the diophantine equation x 2 + 5 k 17 l = y n

A note on the number of $S$ -Diophantine quadruples

Method of infinite ascent applied on $- (2^{p} \cdot A^{6}) + B^{3} = C^{2}$

On the diophantine equation $x^{2} + 5^{k} 17^{l} = y^{n}$