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Keywords:
Pseudoprime; Aurifeuillian pseudoprimes; cyclotomic pseudoprime; strong pseudoprime; superpseudoprimes
Summary:
We use the properties of $p$-adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.
References:
[1] Alford W.R., Granville A., Pomerance C.: There are infinitely many Carmichael numbers. Ann. of Math. 140 (1994), 703–722. DOI 10.2307/2118576 | MR 1283874 | Zbl 0816.11005
[2] Brillhart J., Lehmer D. H., Selfridge L., Tuckerman B., Wagstaff S. S., Jr.: Factorizations of $b^n\pm 1$, $b=2,3,5,6,7,10,11,12$ up to high powers. Contemporary Mathematics, Vol. 22, American Mathematical Society, Providence 1983.
[3] Cipolla M.: Sui numeri composti $P$, che verificano la congruenza di Fermat $\alpha ^{P-1}\equiv 1(\mathop {\rm mod}\,P)$. Annali di Matematica (3) 9 (1904), 139–160. DOI 10.1007/BF02419871
[4] Dickson L. E.: History of the Theory of Numbers. vol. I, New York 1952.
[5] Duparc H. J. A.: Enige generalizaties van de getallen Van Poulet en Carmichael. Math. Centrum Amsterdam, Rapport Z. W. 1956-005.
[6] Erdős P.: On almost primes. Amer. Math. Monthly 57 (1950), 404–407. DOI 10.2307/2307640 | MR 0036259
[7] Granville A. J.: The prime $k$-tuplets conjecture implies that there are arbitrarity long arithmetic progressions of Carmichael numbers. (written communication of December 1995).
[8] Halberstam H., Rotkiewicz A.: A gap theorem for pseudoprimes in arithmetic progression. Acta Arith. 13 (1967/68), 395–404. MR 0225736
[9] Jeans J. A.: The converse of Fermat’s theorem. Messenger of Mathematics 27 (1898), p. 174.
[10] Keller W.: Factors of Fermat numbers and large primes of the form $k\cdot 2^n+1$. Math. Comp., 41 (1983), 661–673. MR 0717710
[11] Keller W.: Prime factors $k\cdot 2^n+1$ of Fermat numbers F_m and complete factoring status of Fermat numbers $F_m$ as of October 5. 2004 URL; http://www.prothsearch.net/fermat.html; Last modified: October 5, 2004.
[12] Kiss E.: Notes on János Bolyai’s researches in number theory. Historia Math. 26 (1999), 68–76. DOI 10.1006/hmat.1998.2212 | MR 1677471 | Zbl 0920.11001
[13] Knopfmacher J., Porubsky: Topologies Related to Arithmetical Properties of Integral Domains. Expo. Math. 15 (1997), 131–148. MR 1458761 | Zbl 0883.11043
[14] Korselt A.: Problème chinois. L’Interm. des Math. 6 (1899), 142-143.
[15] Kraïtchik M.: Théorie des Nombres. Gauthier – Villars, Paris 1922.
[16] Kraïtchik M.: On the factorization of $2^n\pm 1$. Scripta Math. 18 (1952), 39–52.
[17] Křižek M., Luca F., Somer L.: 17 Lectures on Fermat Numbers. From Number Theory to Geometry, Canadian Mathematical Society, Springer 2001. MR 1866957
[18] Lucas E.: Sur la série récurrent de Fermat. Bolletino di Bibliografia e di Storia della Scienze Matematiche e Fisiche 11 (1878), 783–798.
[19] Lucas E.: Théorèmes d’arithmetique. Atti della Reale Accademia delle scienze di Torino 13 (1878), 271–284.
[20] Malo E.: Nombres qui, sans être premiers, vérifient exceptionellement une congruence de Fermat, L’Interm. des Math. 10 (1903), 8.
[21] Mahnke D.: Leibniz and der Suche nach einer allgemeinem Primzahlgleichung. Bibliotheca Math. Vol. 13 (1913), 29–61.
[22] Needham J.: Science and Civilization in China, vol. 3: Mathematics and Sciences of the Heavens and the Earth. Cambridge 1959, p. 54, footnote A. MR 0139507
[23] Pinch Richard G. E.: The pseudoprimes up to $10^{13}$. Algorithmic Number Theory, 4th International Symposium, Proceedings ANTS-IV Leiden, The Netherlands, July 2000, Springer 2000, 456–473. MR 1850626
[24] Pomerance C.: A new lower bound for the pseudoprimes counting function. Illinois J. Math. 26 (1982), 4–9. MR 0638549
[25] Pomerance C., Selfridge J. L., Wagstaff S. S.: The pseudoprimes to $25\cdot 10^9$. Math. Comp. 35 (1980), 1009–1026. MR 0572872
[26] Ribenboim P.: The New Book of Prime Number Records. Springer, New York, 1996. MR 1377060 | Zbl 0856.11001
[27] Riesel H.: Prime Numbers and Computer Methods for Factorization. Birkhäuser, Boston-Basel-Berlin, 1994. MR 1292250 | Zbl 0821.11001
[28] Rotkiewicz A.: Sur les nombres premiers $p$ et $q$ tels que $pq|2^{pq}-2$. Rend. Circ. Mat. Palermo (2) 11 (1962), 280–282. DOI 10.1007/BF02843874 | MR 0166137 | Zbl 0119.03902
[29] Rotkiewicz A.: Sur les nombres pseudopremiers de la forme $ax+b$. C.R. Acad. Sci. Paris 257 (1963), 2601–2604. MR 0162757 | Zbl 0116.03501
[30] Rotkiewicz A., Sierpiński W.: Sur l’équation diophantienne $2^x-xy=2$. Publ. Inst. Math. (Beograd) (N.S.) 4 (18) (1964), 135–137. MR 0171745
[31] Rotkiewicz A., Schinzel A.: Sur les nombres pseudopremiers de la forme $ax^2+bxy+cy^2$. ibidem 258 (1964), 3617–3620. MR 0161828
[32] Rotkiewicz A.: Sur les formules donnant des nombres pseudopremiers. Colloq. Math. 12 (1964), 69–72. MR 0166138 | Zbl 0129.02703
[33] Rotkiewicz A.: Pseudoprime Numbers and Their Generalizations. Stud. Assoc. Fac. Sci. Univ. Novi Sad, 1972, pp. i+169. MR 0330034 | Zbl 0324.10007
[34] Rotkiewicz A.: The solution of W. Sierpiński’s problem. Rend. Circ. Mat. Palermo (2) 28 (1979), 62–64. DOI 10.1007/BF02849586 | MR 0564551 | Zbl 0425.10009
[35] Rotkiewicz A., van der Poorten A. I.: On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc. Ser. A 29 (1980), 316–321. DOI 10.1017/S1446788700021315 | MR 0569519 | Zbl 0428.10001
[36] Sarrus F.: Démonstration de la fausseté du théorème énoncé à la page 320 du $IX^e$ volume de ce recueil. Annales de Math. Pure Appl. 10 (1819–20), 184–187. MR 1556023
[37] Schinzel A.: On primitive prime factors of $a^n-b^n$. Proc. Cambridge Philos. Soc. 58(1962), 555-562. MR 0143728
[38] Sierpiński W.: Remarque sur une hypothèse des Chinois concernant les nombres $(2^n-2)/n$. Colloq. Math. 1 (1948), 9. MR 0023256
[39] Sierpiński W.: A selection of Problems in the Theory of Numbers. Pergamon Press. New York, 1964. MR 0170843
[40] Sierpiński W.: Elementary Theory of numbers. $2^{\rm nd}$ Engl. ed. revised and enlargend by A. Schinzel, Państwowe Wydawnictwo Naukowe, Warszawa, 1988. MR 0930670
[41] Steuerwald R.: Über die Kongruenz $2^{n-1}\equiv 1(\mathop {\rm mod}\,n)$. S.-B. Math.-Nat. Kl., Bayer. Akad. Win., 1947, 177. MR 0030541
[42] Stevenhagen P.: On Aurifeuillian factorizations. Nederl. Akad. Wetensch. Indag. Math. 49 (1987), 451–468. DOI 10.1016/1385-7258(87)90009-6 | MR 0922449 | Zbl 0635.10010
[43] Szymiczek K.: Note on Fermat numbers. Elem. Math. 21 (1966), 598. MR 0193056 | Zbl 0142.28904
[44] Williams Hugh C.: Edouard Lucas and Primality Testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22 A Wiley - Interscience Publication, New York-Chichester-Weinheim-Brisbane-Singapore-Toronto 1998. MR 1632793 | Zbl 1155.11363
[45] Zsigmondy K.: Zur Theorie der Potenzreste. Monastsh. Math. 3 (1892), 265–284. DOI 10.1007/BF01692444 | MR 1546236
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