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Abstract

It is known that the Lucas sequenceV n(ξ,c)=an + bn,a, b being the roots ofx 2 − ξx + c=0 equals the Dickson polynomial\(g_n (\xi ,c) = \sum\limits_{i = 0}^{[n/2]} {\frac{n}{{n - 1}}} \left( {\begin{array}{*{20}c} {n - 1} \\ i \\ \end{array} } \right)( - c)^i \)n−2i Lidl, Müller and Oswald recently defined a number bεℤ to be a strong Dickson pseudoprime to the parameterc (shortlysDpp(c)) if [itgn(b, c)≡b modn for all bεℤ. These numbers seem to be very appropriate for a fast probabilistic prime number test. In generalizing results of the above mentioned authors a criterion is derived for an odd composite number to be ansDpp(c) for fixedc. Furthermore the optimal parameterc for the prime number test is determined.

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References

  1. Lausch, H., Nöbauer, W.: Algebra of Polynomials. Amsterdam, London: North-Holland 1973

    Google Scholar 

  2. Lidl, R., Müller, W. B., Oswald, A.: Some remarks on strong Fibonacci pseudoprimes. Appl. Algebra Engin., Comm. Comp.1 (1990)

  3. Lidl, R., Niederreiter, H.: Finite fields. London, Amsterdam: Addison-Wesley 1983

    Google Scholar 

  4. Müller, W. B., Oswald, A.: On strong Dickson pseudoprimes. Preprint

  5. Nöbauer, W.: Über Permutationspolynome und Permutationsfunktionen für Primzahlpotenzen. Monatsh. Math.69, 230–238 (1965)

    Google Scholar 

  6. Postl, H.: Fast evaluation of Dickson polynomials. In: Contributions to general algebra. Dorninger, D., Eigenthaler, G., Kaiser, H. K., Müller, W. B. (eds.) pp. 223–226. Teubner 1988

  7. Ribenboim, P.: The Book of Prime Number Records. New York, Berlin, Heidelberg: Springer 1988

    Google Scholar 

  8. Rotkiewicz, A.: On the pseudoprimes with respect to the Lucas sequences. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.21, 793–797 (1973)

    Google Scholar 

  9. Williams, H. C.: On numbers analogous to the Carmichael numbers. Canad. Math. Bull.20, 133–143 (1977)

    Google Scholar 

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Kowol, G. On strong Dickson pseudoprimes. AAECC 3, 129–138 (1992). https://doi.org/10.1007/BF01387195

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  • DOI: https://doi.org/10.1007/BF01387195

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