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Unbounded stability of two-term recurrence sequences modulo $2^{k}$

Walter Carlip; Eliot Jacobson — 1996

Acta Arithmetica

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Walter Carlip; Lawrence Somer — 2007

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...

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

Walter Carlip; Lawrence Somer — 2007

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 $U (P, Q)$ such that the rank of $N$ in $U (P, Q)$ is exactly $(N - ε (N)) / d$ , where $ε$ 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.

Symmetry of iteration graphs

Walter Carlip; Martina Mincheva — 2008

Czechoslovak Mathematical Journal

We examine iteration graphs of the squaring function on the rings $ℤ / n ℤ$ 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 $k \geq 5$ and are symmetric when $k = 4$ .

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

Walter Carlip; Lawrence Somer — 2007

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 $p^{r}$ .

Somer, Lawrence; Carlip, Walter — 2000

International Journal of Mathematics and Mathematical Sciences

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Currently displaying 1 – 6 of 6

Unbounded stability of two-term recurrence sequences modulo $2^{k}$

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

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

Symmetry of iteration graphs

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

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

Advanced Search

Formula preview

Currently displaying 1 – 6 of 6

Unbounded stability of two-term recurrence sequences modulo 2 k

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

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

Symmetry of iteration graphs

Bounds for frequencies of residues of second-order recurrences modulo p r

Stability of second-order recurrences modulo p r .

Unbounded stability of two-term recurrence sequences modulo $2^{k}$

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

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

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