Unbounded stability of two-term recurrence sequences modulo
Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers
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 -pseudoprimes and Carmichael-Lucas numbers
Let be a fixed positive integer. A Lucas -pseudoprime is a Lucas pseudoprime for which there exists a Lucas sequence such that the rank of in is exactly , where is the signature of . We prove here that all but a finite number of Lucas -pseudoprimes are square free. We also prove that all but a finite number of Lucas -pseudoprimes are Carmichael-Lucas numbers.
Symmetry of iteration graphs
We examine iteration graphs of the squaring function on the rings when , for a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when and when and are symmetric when .
Bounds for frequencies of residues of second-order recurrences modulo
The authors examine the frequency distribution of second-order recurrence sequences that are not -regular, for an odd prime , and apply their results to compute bounds for the frequencies of -singular elements of -regular second-order recurrences modulo powers of the prime . The authors’ results have application to the -stability of second-order recurrence sequences.
Stability of second-order recurrences modulo .
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