# zbMATH — the first resource for mathematics

On generalized Lucas pseudoprimes. (English) Zbl 0615.10017
Let $$R=R(A,B)=\{R_ n\}^{\infty}_{n=0}$$ be a Lucas sequence of integers defined by $$R_ n=A R_{n-1}-B R_{n-2}$$ $$(n>1)$$, where A and B are integers and the initial terms are $$R_ 0=0$$, $$R_ 1=1$$. We say that R(A,B) is non-degenerate if $$D=A^ 2-4B\neq 0$$ and $$R_ n(A,B)\neq 0$$ for all $$n>0$$. For given non-zero integer D and for any positive integers n, k with $$(n,2D)=1$$ we denote n-k(D/n) by $$\delta _ k(n)$$, where (D/n) is the Jacobi symbol; furthermore let $$n^ *=n/\prod _{p| n}p.$$
The following results are proved. (I) For any given non-zero integer D and positive integers n, k with $$(n,2D)=1$$ and $$n\geq k$$ there exist $$(n^ *,k)\prod _{p| n}\{(\delta _ k(n),\delta _ 1(n))-1\}$$ distinct values of A (mod n), for which there is an integer B such that $$A^ 2-4B\equiv D$$ (mod n) and $$(*)\quad R_{\delta _ k(n)}(A,B)\equiv 0\quad (mod n).$$
(II) For any given non-zero integer D, any positive integer $$k\neq 2$$ there exists infinitely many non-degenerate Lucas sequences R(A,B) such that the congruence (*) has an infinitude of composite solutions n.

##### MSC:
 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B83 Special sequences and polynomials 11A15 Power residues, reciprocity
Full Text:
##### References:
 [1] R. Baillie and S. S. Wagtaff, Jr., Lucas pseudoprimes,Math. Comp.,35 (1980), 1391–1417. [2] Bui Minh Phong, Generalized solution of Rotkiewicz’s problem (in Hungarian),Mat. Lapok (to appear). · Zbl 0748.11010 [3] I. Joó, The number of solutions of the generalized pseudoprime congruencea n n (modn),Ann. Univ. Budapest Eötvös Sect. Math.,31 (1988). [4] P. Kiss and Bui Minh Phong, On a problem of A. Rotkiewicz,Math. Comp. (to appear). · Zbl 0407.10012 [5] D. H. Lehmer, An extended theory of Lucas’ functions,Ann. of Math.,31 (1930), 419–448. · JFM 56.0874.04 [6] A. Makowski, Generalization of Morrow’sD numbers,Simon Stevin,36 (1962), p. 71. · Zbl 0109.27102 [7] W. L. McDaniel, The existence of solutions of the generalized pseudoprime congruencea n n (modn) (to appear). [8] W. L. McDaniel, The generalized pseudoprime congruencea n n (modn),C. R. Math. Rep. Acad. Sci. Canada IX(2) 1987. [9] W. L. McDaniel, A note on the congruencea n n (modn) (to appear). [10] D. C. Morrow, Some properties ofD numbers,Amer. Math. Monthly,58 (1951), 329–330. · Zbl 0043.04302 [11] A. Rotkiewicz,Pseudoprime Numbers and Their Generalizations, Univ. of Novi Sad, 1972. · Zbl 0324.10007 [12] A. Rotkiewicz, On the congruence 2 n (modn),Math. Comp.,43 (1984), 271–272. · Zbl 0542.10003 [13] M. K. Shen, On the congruence 2 n (modn),Math. Comp.,46 (1986), 715–716. · Zbl 0588.10002 [14] H. C. Williams, On the numbers analogous to the Carmichael numbers,Canad. Math. Bull.,20 (1977), 133–143. · Zbl 0368.10011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.