On some special forms of simultaneous Pell equations

Zhigang Li; Jianye Xia; Pingzhi Yuan

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On the number of nonquadratic residues which are not primitive roots

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We show that there exist infinitely many positive integers r not of the form (p-1)/2 - ϕ(p-1), thus providing an affirmative answer to a question of Neville Robbins.

Primitive Lucas d-pseudoprimes and Carmichael-Lucas numbers

Walter Carlip, Lawrence Somer (2007)

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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. ...