Abstract
Following recent papers [1], [5], [22], [23] a composite n is called a pseudoprime to base \(b if {{b}^{{n - 1}}} \equiv 1 \bmod n. \)
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References
Alford, W.R., Granville, A. and Pomerance C. “There are infinitely many Carmichael numbers.” Ann. of Math., Vol. 140 (1994): pp. 703–722.
Baillie R. and Wagstaff Samuel S. “Lucas pseudoprimes.” Math. Comp., Vol. 35 (1980): pp. 1391–1417.
Benkoski, S.J. Review of On the congruence \({{{\text{2}}}^{{n - k}}} \equiv \left( {\bmod n} \right). \). MR87e: 11005.
Cipolla, M. “Sui numeri composti P che verificaro la congruenza di Fermat \({{a}^{{P - 1}}} \equiv 1\left( {\bmod P} \right). \).” Annali di Matematica (3), Vol. 9 (1904): pp. 139–160.
Conway, J.H., Guy, R.K., Schneeberger, W.A. and Sloane, N.J.A. “The primary pretenders.” Acta Arith., Vol. 78 (1997): pp. 307–313.
Dickson, L.E. History of the Theory of Numbers. Carnegie Institute, Washington, 1919, 1920, 1923; reprinted Stechert, New York, 1934; Chelsea, New York, 1952, 1960, Vol. I, Chap. III.
Erdös, P., Kiss, P. and Sárközy, A. “Lower bound for the counting function.” Math. Comp., Vol. 51 (1988): pp. 315–323.
Fehér, J. and Kiss, P. “Note on super pseudoprime numbers.” Ann. Univ. Sci. Budapest, Eötvös Sect. Math., Vol. 26 (1983): pp. 157–159.
Grantham, J.F. Frobenius pseudoprimes. A dissertation submitted to the Graduate Faculty of the University of Georgia, Athens GA 1997.
Granville, A. The prime k-tuplets conjecture implies that there are arbitrarily long arithmetic progressions of Carmichael numbers, (written communication).
Guy, R.K. Unsolved Problems in Number Theory. Springer-Verlag, New York-Heidelberg, XVI + 285, p.p. 1994.
Kiss, P. and Phong, B.M. “On a problem of A. Rotkiewicz.” Math. Comp., Vol. 48 (1987): pp. 751–755.
Korselt, A. “Probléme chinois.” L’intermédaire des mathématiciens, Vol. 6 (1899): pp. 142–143.
Lehmer E. “On the infinitude of Fibonacci pseudoprimes.” The Fibonacci Quarterly, Vol. 2 (1964): pp. 229–230.
Mahnke, D. “Leibniz and der Suche nach einer allgemeinem Primzahlgleichung.” Bibliotheca Math., Vol. 13 (1913): pp. 29–61.
Ma¸kowski, A. “Generalization of Morrow’s D numbers.” Simon Stevin, Vol. 36 (1962): p. 71.
McDaniel, W.L. “The generalized pseudoprime congruence \({{a}^{{n - k}}} \equiv {{b}^{{n - k}}}\left( {\bmod n} \right). \).” C. R. Math. Rep. Acad. Sci. Canada, Vol. 9 (1987): pp. 143–148.
McDaniel, W.L. “Some pseudoprimes and related numbers having special forms.” Math. Comp., Vol. 53 (1989): pp. 407–409.
McDaniel, W.L. “The existence of solutions of the generalized pseudoprime congruence \({{a}^{{f(n)}}} \equiv {{b}^{{f(n)}}}\left( {\bmod n} \right). \).” Colloq. Math., Vol. 49 (1990): pp. 177–190.
Morrow, D.C. “Some properties of D numbers.” Amer. Math. Monthly, Vol. 58 (1951): pp. 329–330.
Phong, B.M. “Generalized solutions of Rotkiewicz’s problem.” Matematikai Lapok, Vol. 34(1–3) (1987): pp. 109–119.
Pomerance, C. “A new lower bound for the pseudoprimes counting function.” Illinois J. Math., Vol. 26 (1982): pp. 4–9; MR 83h: 1012.
Pomerance, C., Selfridge, I.L. and Wagstaff, Samuel S. “The pseudoprimes to 25 · 109.” Math. Comp., Vol. 35 (1980): pp. 1003–1026.
Ribenboim, P. The New Book of Prime Number Records, Springer-Verlag, New York-Berlin-Heidelberg, 1995.
Rotkiewicz, A. “Sur les nombres composés n qui divisent a n-1-b n-1.” Rend. Circ. Mat. Palermo (2), Vol. 8 (1959): pp. 115–116; MR. 23#A1579.
Rotkiewicz, A. “Sur quelques généralisations des nombres pseudopremiers.” Colloq. Math., Vol. 9 (1962): pp. 109–113.
Rotkiewicz, A. “Sur les nombres premiers p et q tels que pq | 2pq-2.” Rend Circ. Mat. Palermo, Vol. 11 (1962): pp. 280–282.
Rotkiewicz, A. “Sur les nombres pseudopremiers de la forme ax + b.” C. R. Acad. Sci. Paris, Vol. 257 (1963): pp. 2601–2604.
Rotkiewicz, A. “Sur progressions arithmétiques et gómétriques formés de trois nombres pseudopremiers distincts.” Acta Arith., Vol. 10 (1964): pp. 325–328.
Rotkiewicz, A. “Sur les nombres pseudopremiers triangulaires.” Elem. Math., Vol. 19 (1964): pp. 82–83.
Rotkiewicz, A. “Sur les nombres pseudopremiers pentagonaux.” Bull. Soc. Roy. Sci. Liége, Vol. 33 (1964): pp. 261–263.
Rotkiewicz, A. “On the pseudoprimes of the form ax + b.” Proc. Combridge Phil. Soc., Vol. 63 (1963): pp. 389–392.
Rotkiewicz, A. “On the prime factors of the number 2p-1-1.” Glasgow Math. J., Vol. 9 (1968): pp. 83–86.
Rotkiewicz, A. “On arithmetical progressions formed by k different pseudoprimes.” Journal of Math. Sciences, Vol. 4 (1969): pp. 5–10.
Rotkiewicz, A. “Pseudoprime Numbers and Their Generalizations.” Student Association of Faculty of Sciences, Univ. of Novi Sad, 1972, i + 169, pp.
Rotkiewicz, A. “The solution of W. Sierpiński’s problem.” Rend. Circ. Mat. Palermo, Vol. 28 (1979): pp. 62–64.
Rotkiewicz, A. “Arithmetical progressions formed from three different Euler pseudoprimes for the odd base a.” Rend. Circ. Mat. Palermo, Vol. 29 (1980): pp. 420–426.
Rotkiewicz, A. “On the congruence \({{2}^{{n - 2}}} \equiv 1\left( {\bmod n} \right). \).” Math. Comp., Vol. 43 (1984): pp. 271–272.
Rotkiewicz, A. “Arithmetical progressions formed by k different pseudoprimes.” Rend. Circ. Mat. Palermo, Vol. 69 (1994).
Rotkiewicz, A. “There are infinitely many arithmetical progressions formed by three different Fibonacci pseudoprimes.” Applications of Fibonacci Numbers. Volume 7. Edited by G.E. Bergum, A.F. Horadam and A.N. Philippou. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998: pp. 327–332.
Rotkiewicz, A. “Arithmetical progressions formed by Lucas pseudoprimes.” Number Theory, Proceedings Eger Conference. Edited by K. Györy, A. Pethö, V.T. Sós, Walter de Gruyter. Berlin-New York 1998: pp. 465–472.
Rotkiewicz, A. “Lucas and Lehmer pseudoprimes with negative Jacobi symbol.” (to appear in Acta Arith.).
Rotkiewicz, A. “Periodic sequences of pseudoprimes connected with Carmichael numbers and the least period of the function l x C.” (to appear in Acta Anth.).
Schinzel, A. “Sur les nombres composés n qui divisent a n-a.” Rend. Circ. Mat. Palermo (2), Vol. 7 (1958): pp. 37–41.
Schinzel, A. and Sierpiński, W. “Sur certaines hypothèses concernnt les nombres premiers.” Acta Arith., Vol. 4 (1958): pp. 185–208, Correction ibid. Vol. 5 (1958): p. 259.
Shen, Mok-King. “On the congruence \({{2}^{{n - 2}}} \equiv 1\left( {\bmod n} \right). \).” Math. Comp., Vol. 46 (1986): pp. 715–716.
Sierpiński, W. “Remark on composite numbers m dividing a m-a (in Polish).” Wiadom. Mat, Vol. 4 (1961): pp. 183–184.
Sierpiński, W. “Elementary Theory of Numbers.” Monografie Matematyczne, Vol. 42 PWN, Warsaw (1964), (second edition, North-Holland, Amsterdam, 1987).
Williams, H.C. “On numbers analogous to the Carmichael numbers.” Canad. Math. Bull., Vol. 20.1 (1977): pp. 133–143.
Zhang, Ming-Zhi. “A note on the congruence \({{{\text{2}}}^{{n - 2}}} \equiv \bmod n \).” (Chinese and English summary), Sichuan Daxue Xuebao, Vol. 27 (1990): pp. 130–131.
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Rotkiewicz, A. (1999). Solved and Unsolved Problems on Pseudoprime Numbers and Their Generalizations. In: Howard, F.T. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4271-7_28
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DOI: https://doi.org/10.1007/978-94-011-4271-7_28
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