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2. Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers
- Creator:
- Carlip, Walter and Somer, Lawrence
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- Lucas, Fibonacci, pseudoprime, and Fermat
- Language:
- English
- Description:
- Let $d$ be a fixed positive integer. A Lucas $d$-pseudoprime is a Lucas pseudoprime $N$ for which there exists a Lucas sequence $U(P,Q)$ such that the rank of $N$ in $U(P,Q)$ is exactly $(N - \varepsilon (N))/d$, where $\varepsilon $ is the signature of $U(P,Q)$. We prove here that all but a finite number of Lucas $d$-pseudoprimes are square free. We also prove that all but a finite number of Lucas $d$-pseudoprimes are Carmichael-Lucas numbers.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Symmetry of iteration graphs
- Creator:
- Carlip, Walter and Mincheva, Martina
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- digraph, iteration digrap, quadratic map, cycle, and tree
- Language:
- English
- Description:
- We examine iteration graphs of the squaring function on the rings $\mathbb{Z}/n\mathbb{Z}$ when $n = 2^{k}p$, for $p$ a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when $k=3$ and when $k\ge 5$ and are symmetric when $k = 4$.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public