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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...

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 ≥ 5 and are symmetric when k = 4 .

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.

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.

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