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Acta Arithmetica

### Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Colloquium Mathematicae

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 appearance of N in U(P,Q) is exactly (N-ε(N))/d, where the signature ε(N) = (D/N) is given by the Jacobi symbol with respect to the discriminant D of U. A Lucas d-pseudoprime N is a primitive Lucas d-pseudoprime if (N-ε(N))/d is the maximal rank of N among Lucas sequences U(P,Q) that exhibit N as a Lucas pseudoprime. We derive...

### Symmetry of iteration graphs

Czechoslovak Mathematical Journal

We examine iteration graphs of the squaring function on the rings $ℤ/nℤ$ 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 $k=4$.

### Square-free Lucas $d$-pseudoprimes and Carmichael-Lucas numbers

Czechoslovak Mathematical Journal

Let $d$ be a fixed positive integer. A Lucas $d$-pseudoprime is a Lucas pseudoprime $N$ for which there exists a Lucas sequence $U\left(P,Q\right)$ such that the rank of $N$ in $U\left(P,Q\right)$ is exactly $\left(N-\epsilon \left(N\right)\right)/d$, where $\epsilon$ is the signature of $U\left(P,Q\right)$. 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.

### Bounds for frequencies of residues of second-order recurrences modulo ${p}^{r}$

Mathematica Bohemica

The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences.

### Stability of second-order recurrences modulo ${p}^{r}$.

International Journal of Mathematics and Mathematical Sciences

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