Lucas and Frobenius pseudoprimes
Abstract
We define several types of Lucas and Frobenius pseudoprimes and prove some theorems on these pseudoprimes. In particular: There exist infinitely many arithmetic progressions formed by three different Frobenius-Fibonacci pseudoprimes.
References
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9. J.F. Grantham, Frobenius pseudoprimes, University of Georgia, Athens GA. 1997 (dis sertation).
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13. R.G.E. Pinch, The pseudoprimes up to 10^{13} , Algorithmic Number Theory, 4th International Symposium, ANTS-IV Leiden, The Netherlands, 2-7 July 2000, Proceedings (2000), 459-473.
14. C. Pomerance, J.L. Selfridge, S.S. Wagstaff, The pseudoprimes to 25•10^9, Math. of Comput. 35 (1980), 1003-1026.
15. P. Poulet, Table des nombres composés vérifiant le théoreme de Fermat pour le module 2 jusqu'à 100.000.000, Sphinx 8 (1938), 42-52. Errata in Math. of Comput. 25 (1971), 944-945, Math. of Comput. 26 (1972), 814.
16. P. Ribenboim, The new Book of Prime Number Records, Springer, New York 1996.
17. A. Rotkiewicz, On the pseudoprimes of the form ax+b with respect to the sequence of Lehmer, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 20 (1972), 83-85.
18. A. Rotkiewicz, On the pseudoprimes with respect to the Lucas sequence, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 21 (1973), 793-797.
19. A. Rotkiewicz, On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters L, Q in arithmetic progression, Math. of Comput. 39 (1982), 239-247.
20. A. Rotkiewicz, On strong Lehmer pseudoprimes in the case, of negative discriminant in arithmetic progressions, Acta Arithm. 68 (1994), 145-151.
21. A. Rotkiewicz, There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes, Applications of Fibonacci Numbers, Vol. 7, Ed. by G.E. Bergum, A.N. Philippou and A.F. Horadam. Kluwer Academic Publishers, Dordrecht, the Netherlands 1998, 327-332.
22. A. Rotkiewicz, Arithmetical progression formed by Lucas pseudoprimes, in: Number Theory, Diophantine, Computational and Algebraic Aspects, Eds: Győrgy Kálmán, Attila Pethő and Vera T. Sós. Walter de Gruyter GmbH & Co., Berlin, New York 1998, pp. 465-472.
23. A. Rotkiewicz, Lucas pseudoprimes, Funct. Approximatio Comment. Math. 28 (2000), 97-104.
24. A. Rotkiewicz, A. Schinzel, Lucas pseudoprimes with a prescribed value of the Jacobi symbol, Bull. Polish Acad. Sci. Math. 48 (2000), 77-80.
25. A. Rotkiewicz, K. Ziemak, On even pseudoprimes, The Fibonacci Quarl. 33 (1995), 123-125.
26. M. Yorinaga, On a congruental property of Fibonacci numbers. Numerical experiments. Considerations and Remarks, Math. J. Okayama Univ. 19 (1976), 5-10, 11-17.
27. Z. Zhang, Finding strong pseudoprimes to several bases, Math. of Comput. 70 (2000), 863-872.
2. T. Banachiewicz, O związku pomiędzy pewnym twierdzeniem matematyków chińskich a formą Fermata na liczby pierwsze, Spraw. Tow. Nauk., Warszawa, 2 (1909), 7-11.
3. N.G.W.H. Beeger, On even numbers m dividing 2^m-2, Amer. Math. Monthly 58 (1951), 553-555.
4. R.E. Crandall, C. Pomerance, Prime Numbers. A Computational Perspective, Springer, New York 2001.
5. L.E. Dickson, History of the Theory of Numbers, vol. I: Divisibility and Primality, Carnegie Inst., Washington 1919.
6. K. Dilcher, Fermat numbers, Wieferich and Wilson primes: Computations and generations, Proc. Conf. on Computational Number Theory and Public Key Cryptography, Warsaw. Sept. 2000, 1-22.
7. P. Erdős, On almost prunes, Amer. Math. Monthly 57 (1950), 404-407.
8. P. Erdős, P. Kiss, A. Sárközy, Lower bound for the counting function of Lucas pseudoprimes, Math. of Comput. 51 (1988), 315-323.
9. J.F. Grantham, Frobenius pseudoprimes, University of Georgia, Athens GA. 1997 (dis sertation).
10. J.F. Grantham, Frobenius pseudoprimes, Math. of Comput. 70 (2001), 871-891.
11. R.K. Guy, Unsolved Problems in Number Theory, Springer-Verlag, New York 1994.
12. M. Křížek, F. Luca, L. Somer, 17 Lectures on Fermat Numbers, From Number Theory to Geometry, Springer-Verlag, New York 2001.
13. R.G.E. Pinch, The pseudoprimes up to 10^{13} , Algorithmic Number Theory, 4th International Symposium, ANTS-IV Leiden, The Netherlands, 2-7 July 2000, Proceedings (2000), 459-473.
14. C. Pomerance, J.L. Selfridge, S.S. Wagstaff, The pseudoprimes to 25•10^9, Math. of Comput. 35 (1980), 1003-1026.
15. P. Poulet, Table des nombres composés vérifiant le théoreme de Fermat pour le module 2 jusqu'à 100.000.000, Sphinx 8 (1938), 42-52. Errata in Math. of Comput. 25 (1971), 944-945, Math. of Comput. 26 (1972), 814.
16. P. Ribenboim, The new Book of Prime Number Records, Springer, New York 1996.
17. A. Rotkiewicz, On the pseudoprimes of the form ax+b with respect to the sequence of Lehmer, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 20 (1972), 83-85.
18. A. Rotkiewicz, On the pseudoprimes with respect to the Lucas sequence, Bull. Acad. Polon. Sci. Sér. Math. Astronom. Phys. 21 (1973), 793-797.
19. A. Rotkiewicz, On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters L, Q in arithmetic progression, Math. of Comput. 39 (1982), 239-247.
20. A. Rotkiewicz, On strong Lehmer pseudoprimes in the case, of negative discriminant in arithmetic progressions, Acta Arithm. 68 (1994), 145-151.
21. A. Rotkiewicz, There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes, Applications of Fibonacci Numbers, Vol. 7, Ed. by G.E. Bergum, A.N. Philippou and A.F. Horadam. Kluwer Academic Publishers, Dordrecht, the Netherlands 1998, 327-332.
22. A. Rotkiewicz, Arithmetical progression formed by Lucas pseudoprimes, in: Number Theory, Diophantine, Computational and Algebraic Aspects, Eds: Győrgy Kálmán, Attila Pethő and Vera T. Sós. Walter de Gruyter GmbH & Co., Berlin, New York 1998, pp. 465-472.
23. A. Rotkiewicz, Lucas pseudoprimes, Funct. Approximatio Comment. Math. 28 (2000), 97-104.
24. A. Rotkiewicz, A. Schinzel, Lucas pseudoprimes with a prescribed value of the Jacobi symbol, Bull. Polish Acad. Sci. Math. 48 (2000), 77-80.
25. A. Rotkiewicz, K. Ziemak, On even pseudoprimes, The Fibonacci Quarl. 33 (1995), 123-125.
26. M. Yorinaga, On a congruental property of Fibonacci numbers. Numerical experiments. Considerations and Remarks, Math. J. Okayama Univ. 19 (1976), 5-10, 11-17.
27. Z. Zhang, Finding strong pseudoprimes to several bases, Math. of Comput. 70 (2000), 863-872.
RotkiewiczA. (2003). Lucas and Frobenius pseudoprimes. Annales Mathematicae Silesianae, 17, 17-39. Retrieved from https://www.journals.us.edu.pl/index.php/AMSIL/article/view/14098
Andrzej Rotkiewicz
rotkiewi@impan.gov.pl
Instytut Matematyki, Polska Akademia Nauk Poland
Instytut Matematyki, Polska Akademia Nauk Poland
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