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AN UPPER BOUND FOR THE NUMBER OF DIOPHANTINE QUINTUPLES
- Part of
- Journal: Bulletin of the Australian Mathematical Society / Volume 94 / Issue 3 / December 2016
- Published online by Cambridge University Press: 16 August 2016, pp. 384-394
- Print publication: December 2016
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We improve the known upper bound for the number of Diophantine
$D(4)$-quintuples by using the most recent methods that were developed in the 
$D(1)$ case. More precisely, we prove that there are at most 
$6.8587\times 10^{29}$
$D(4)$-quintuples.
Jeśmanowicz’ Conjecture with Congruence Relations. II
- Journal: Canadian Mathematical Bulletin / Volume 57 / Issue 3 / 01 September 2014
- Published online by Cambridge University Press: 20 November 2018, pp. 495-505
- Print publication: 01 September 2014
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Let
$a$, 
$b$, and 
$c$ be primitive Pythagorean numbers such that 
${{a}^{2}}\,+\,{{b}^{2}}\,=\,{{c}^{2}}$ with $b$ even. In this paper, we show that if 
${{b}_{0}}\,\equiv \,\in \,\,\,\left( \bmod \,a \right)$ with 
$\text{ }\!\!\varepsilon\!\!\text{ }\,\in \,\left\{ \pm 1 \right\}$ for certain positive divisors 
${{b}_{0}}$ of $b$, then the Diophantine equation 
${{a}^{x}}\,+\,{{b}^{y}}\,=\,{{c}^{z}}$ has only the positive solution 
$\left( x,\,y,\,z \right)\,=\,\left( 2,\,2,\,2 \right)$.
Sign Changes of the Liouville Function on Quadratics
- Journal: Canadian Mathematical Bulletin / Volume 56 / Issue 2 / 01 June 2013
- Published online by Cambridge University Press: 20 November 2018, pp. 251-257
- Print publication: 01 June 2013
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Let
$\text{ }\!\!\lambda\!\!\text{ }\left( n \right)$ denote the Liouville function. Complementary to the prime number theorem, Chowla
conjectured that*
$$\sum\limits_{n\le x}{\lambda \,\left( f\left( n \right) \right)}=o\left( x \right)$$for any polynomial
$f\left( x \right)$ with integer coefficients which is not of form 
$bg{{\left( x \right)}^{2}}$.
