Pseudoprvočísla

Křížek, Michal   ; Somer, Lawrence

Pokroky matematiky, fyziky a astronomie, Tome 48 (2003) no. 2, pp. 143-151

Détail

Classification : 11-xx, 11A51
Mots-clés : Fermat theorem; prime; Carmichael number

Bibliographie

[1] Alford, W. R., Granville, A., Pomerance, C.: There are infinitely many Carmichael numbers. Ann. of Math. 140 (1994), 703–722. | DOI | MR | Zbl

[2] Baillie, R., Wagstaff, S. S.: Lucas pseudoprimes. Math. Comp. 35 (1980), 1391–1417. | DOI | MR | Zbl

[3] Banachiewicz, T.: O związku pomiędzy pewnym twierdzeniem matematyków chińskich a formą Fermata na liczby pierwsze. Spraw. Tow. Nauk, Warszawa 2 (1909), 7–11.

[4] Beeger, N. G. W. H.: On even numbers $m$ dividing ${2^m-2}$. Amer. Math. Monthly 58 (1951), 553–555. | DOI | MR

[5] Carmichael, R. D.: Note on a new number theory function. Bull. Amer. Math. Soc. 16 (1910), 232–238. | DOI | MR

[6] Carmichael, R. D.: On composite numbers $P$ which satisfy the Fermat congruence ${a^{P-1}\equiv 1 (\operatorname{mod} P)}$. Amer. Math. Monthly 19 (1912), 22–27. | DOI | MR

[7] Cipolla, M.: Sui numeri composti P, che verificano la congruenza di Fermat ${a^{P-1}\equiv 1 (\operatorname{mod} P)}$. Annali di Matematica (3) 9 (1904), 139–160. | DOI

[8] Dickson, L. E.: History of the theory of numbers, vol. I, Divisibility and primality. Carnegie Inst., Washington 1919.

[9] Duparc, H. J. A.: On Carmichael numbers, Poulet numbers, Mersenne numbers and Fermat numbers. Rapport ZW 1953-004, Math. Centrum Amsterdam 1953, 1–7. | MR | Zbl

[10] Erdős, P.: On almost primes. Amer. Math. Monthly 57 (1950), 404–407. | DOI | MR

[11] Jarden, D.: Existence of an infinitude of composite $n$ for which ${2^{n-1}\equiv 1 (\operatorname{mod} n)}$ (Hebrew, Engl. Summary). Riveon Lematematika 4 (1950), 65–67. | MR

[12] Jeans, J. H.: The converse of Fermat’s theorem. Messenger of Mathematics 27 (1897/98), 174.

[13] Joo, I., Phong, B. M.: On super Lehmer pseudoprimes. Studia Sci. Math. Hungar. 25 (1990), 121–124. | MR

[14] Kiss, E.: Notes on János Bolyai’s researches in number theory. Historia Math. 26 (1999), 68–76. | DOI | MR | Zbl

[15] Kiss, P.: Some results on Lucas pseudoprimes. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 28 (1985), 153–159. | MR

[16] Korselt, A.: Problème chinois. L’Interm. des Math. 6 (1899), 143.

[17] Křížek, M., Luca, F., Somer, L.: 17 lectures on Fermat numbers: From number theory to geometry. CMS Books in Mathematics, vol. 9, Springer-Verlag, New York 2001. | MR | Zbl

[18] Lehmer, D. H.: On the converse of Fermat’s theorem. Amer. Math. Monthly 43 (1936), 346–354. | DOI | MR | Zbl

[19] Mahnke, D.: Leibniz auf der Suche nach einer allgemeinen Primzahlgleichung. Bibliotheca Math. 13 (1913), 29–61.

[20] Mąkowski, A.: On a problem of Rotkiewicz on pseudoprime numbers. Elem. Math. 29 (1974), 13. | MR

[21] Malo, E.: Nombres qui sans être premiers, verifient exceptionnellement une congruence de Fermat. L’Interm. des Math. 10 (1903), 88.

[22] Phong, B. M.: On super Lucas and super Lehmer pseudoprimes. Studia Sci. Math. Hungar. 23 (1988), 435–442. | MR | Zbl

[23] Pomerance, C.: On the distribution of pseudoprimes. Math. Comp. 37 (1981), 587–593. | DOI | MR | Zbl

[24] Pomerance, C.: A new lower bound for the pseudoprime counting function. Illinois J. Math. 26 (1982), 4–9. | MR | Zbl

[25] Pomerance, C., Selfridge, J. L., Wagstaff, S. S.: The pseudoprimes to ${25\cdot 10^9}$. Math. Comp. 35 (1980), 1003–1026. | DOI | MR

[26] Porubský, Š.: Fermat a teorie čísel aneb Problematika dělitelů a dokonalá čísla. In: Matematik Pierre de Fermat (eds. A. Šolcová et al.), Cahiers du CEFRES 28 (2002), 49–86.

[27] Poulet, P.: Table des nombres composés vérifiant le théeorème de Fermat pour le module $2$ jusqu’à $100.000.000$. Sphinx 8 (1938), 42–52. Errata in Math. Comp. 25 (1971), 944–945, Math. Comp. 26 (1972), 814. | MR

[28] Ribenboim, P.: The book of prime number records. Springer, New York 1988, 1989. | MR | Zbl

[29] Ribenboim, P.: The new book of prime number records. Springer, New York 1996. | MR | Zbl

[30] Rotkiewicz, A.: Sur les nombres pseudopremiers de la forme ${ax+b}$. C. R. Acad. Sci. Paris Sér. I Math. 257 (1963), 2601–2604. | MR | Zbl

[31] Rotkiewicz, A.: Sur les formules donnant des nombres pseudopremiers. Colloq. Math. 12 (1964), 69–72. | MR | Zbl

[32] Rotkiewicz, A.: On the pseudoprimes of the form ${ax+b}$. Proc. Cambridge Philos. Soc. 63 (1967), 389–392. | MR | Zbl

[33] Rotkiewicz, A.: Pseudoprime numbers and their generalizations. Stud. Assoc. Fac. Sci. Univ. Novi Sad 1972. | MR | Zbl

[34] Rotkiewicz, A.: Lucas and Frobenius pseudoprimes. Proc. of the 10th Internat. Conf. on Fibonacci Numbers and their Applications, Flagstaff, Arizona, 2002 (to appear in Kluwer), 1–21. | MR

[35] Sierpiński, W.: Remarque sur une hypothèse des Chinois concernant les nombres ${(2^n-2)/n}$. Colloq. Math. 1 (1948), 9. | MR

[36] Somer, L.: On Fermat $d$-pseudoprimes. In: Théorie des nombres (éd. J.-M. De Koninck, C. Levesque), Walter de Gruyter, Berlin, New York, 1989, 841–860. | MR | Zbl

[37] Somer, L.: On Lucas $d$-pseudoprimes. In: Applications of Fibonacci numbers, vol. 7 (eds. G. E. Bergum, A. N. Philippou, A. F. Horadam), Kluwer Academic Publishers, Dordrecht 1998, 369–375. | MR | Zbl

[38] Steuerwald, R.: Über die Kongruenz ${2^{n-1}\equiv 1 (\operatorname{mod} n)}$. S.-B. Math.-Nat. Kl., Bayer. Akad. Wiss. 1947, 177. | MR

[39] Szymiczek, K.: Note on Fermat numbers. Elem. Math. 21 (1966), 59. | MR | Zbl

[40] Szymiczek, K.: On pseudoprimes which are products of distinct primes. Amer. Math. Monthly 74 (1967), 35–37. | DOI | MR | Zbl