Contents
Research Article
GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS
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- 21 December 2018, pp. 272-288
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We study generalized continued fraction expansions of the form
where
$$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$
$N$ is a fixed positive integer and the partial numerators 
$a_{i}$ are positive integers for all 
$i$. We call these expansions 
$\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many 
$\operatorname{dn}_{N}$ expansions for each 
$N$. In particular, we study the 
$\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each 
$N$, a 
$\operatorname{dn}_{N}$ expansion with bounded partial numerators.
Front Cover (OFC, IFC) and matter
JAZ volume 107 Issue 2 Cover and Front matter
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- Published online by Cambridge University Press:
- 09 September 2019, pp. f1-f2
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JAZ volume 107 Issue 3 Cover and Front matter
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- Published online by Cambridge University Press:
- 21 November 2019, pp. f1-f2
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Research Article
ON VALUES TAKEN BY THE LARGEST PRIME FACTOR OF SHIFTED PRIMES
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- 15 August 2018, pp. 133-144
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Denote by
$\mathbb{P}$ the set of all prime numbers and by 
$P(n)$ the largest prime factor of positive integer 
$n\geq 1$ with the convention 
$P(1)=1$. In this paper, we prove that, for each 
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant 
$c(\unicode[STIX]{x1D702})>1$ such that, for every fixed nonzero integer 
$a\in \mathbb{Z}^{\ast }$, the set has relative asymptotic density one in
$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}<q\leq c(\unicode[STIX]{x1D702})p^{\unicode[STIX]{x1D702}}\}\end{eqnarray}$$
$\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’, J. Aust. Math. Soc.82 (2015), 133–147], Theorem 1.1, which requires 
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$ in place of 
$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.
Back Cover (OBC, IBC) and matter
JAZ volume 107 Issue 2 Cover and Back matter
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- Published online by Cambridge University Press:
- 09 September 2019, pp. b1-b2
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JAZ volume 107 Issue 1 Cover and Front matter
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- Published online by Cambridge University Press:
- 05 July 2019, pp. f1-f2
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JAZ volume 107 Issue 3 Cover and Back matter
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- Published online by Cambridge University Press:
- 21 November 2019, pp. b1-b7
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JAZ volume 107 Issue 1 Cover and Back matter
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- Published online by Cambridge University Press:
- 05 July 2019, pp. b1-b2
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