On pseudoprimes having special forms and a solution of K. Szymiczek’s problem

Andrzej Rotkiewicz

Acta Mathematica Universitatis Ostraviensis (2005)

  • Volume: 13, Issue: 1, page 57-71
  • ISSN: 1804-1388

Abstract

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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.

How to cite

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Rotkiewicz, Andrzej. "On pseudoprimes having special forms and a solution of K. Szymiczek’s problem." Acta Mathematica Universitatis Ostraviensis 13.1 (2005): 57-71. <http://eudml.org/doc/35153>.

@article{Rotkiewicz2005,
abstract = {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.},
author = {Rotkiewicz, Andrzej},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {Pseudoprime; Aurifeuillian pseudoprimes; cyclotomic pseudoprime; strong pseudoprime; superpseudoprimes; pseudoprimes; Aurifeuillian pseudoprimes; cyclotomic pseudoprime; strong pseudoprime; superpseudoprimes},
language = {eng},
number = {1},
pages = {57-71},
publisher = {University of Ostrava},
title = {On pseudoprimes having special forms and a solution of K. Szymiczek’s problem},
url = {http://eudml.org/doc/35153},
volume = {13},
year = {2005},
}

TY - JOUR
AU - Rotkiewicz, Andrzej
TI - On pseudoprimes having special forms and a solution of K. Szymiczek’s problem
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2005
PB - University of Ostrava
VL - 13
IS - 1
SP - 57
EP - 71
AB - 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.
LA - eng
KW - Pseudoprime; Aurifeuillian pseudoprimes; cyclotomic pseudoprime; strong pseudoprime; superpseudoprimes; pseudoprimes; Aurifeuillian pseudoprimes; cyclotomic pseudoprime; strong pseudoprime; superpseudoprimes
UR - http://eudml.org/doc/35153
ER -

References

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  1. Alford W.R., Granville A., Pomerance C., 10.2307/2118576, , Ann. of Math. 140 (1994), 703–722. (1994) Zbl0816.11005MR1283874DOI10.2307/2118576
  2. Brillhart J., Lehmer D. H., Selfridge L., Tuckerman B., Wagstaff S. S., Jr., Factorizations of b n ± 1 , b = 2 , 3 , 5 , 6 , 7 , 10 , 11 , 12 up to high powers, , Contemporary Mathematics, Vol. 22, American Mathematical Society, Providence 1983. (1983) 
  3. Cipolla M., 10.1007/BF02419871, , Annali di Matematica (3) 9 (1904), 139–160. (1904) DOI10.1007/BF02419871
  4. Dickson L. E., History of the Theory of Numbers, , vol. I, New York 1952. (1952) 
  5. Duparc H. J. A., Enige generalizaties van de getallen Van Poulet en Carmichael, , Math. Centrum Amsterdam, Rapport Z. W. 1956-005. (1956) 
  6. Erdős P., 10.2307/2307640, , Amer. Math. Monthly 57 (1950), 404–407. (1950) MR0036259DOI10.2307/2307640
  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). (1995) 
  8. Halberstam H., Rotkiewicz A., A gap theorem for pseudoprimes in arithmetic progression, , Acta Arith. 13 (1967/68), 395–404. (1967) MR0225736
  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 · 2 n + 1 , , Math. Comp., 41 (1983), 661–673. (1983) MR0717710
  11. Keller W., Prime factors k · 2 n + 1 of Fermat numbers Fm 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., 10.1006/hmat.1998.2212, , Historia Math. 26 (1999), 68–76. (1999) Zbl0920.11001MR1677471DOI10.1006/hmat.1998.2212
  13. Knopfmacher J., Porubsky, Topologies Related to Arithmetical Properties of Integral Domains, , Expo. Math. 15 (1997), 131–148. (1997) Zbl0883.11043MR1458761
  14. Korselt A., Problème chinois, , L’Interm. des Math. 6 (1899), 142-143. 
  15. Kraïtchik M., Théorie des Nombres, , Gauthier – Villars, Paris 1922. (1922) 
  16. Kraïtchik M., On the factorization of 2 n ± 1 , , Scripta Math. 18 (1952), 39–52. (1952) 
  17. Křižek M., Luca F., Somer L., 17 Lectures on Fermat Numbers, , From Number Theory to Geometry, Canadian Mathematical Society, Springer 2001. MR1866957
  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. (1903) 
  21. Mahnke D., Leibniz and der Suche nach einer allgemeinem Primzahlgleichung, , Bibliotheca Math. Vol. 13 (1913), 29–61. (1913) 
  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. (1959) MR0139507
  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. MR1850626
  24. Pomerance C., A new lower bound for the pseudoprimes counting function, , Illinois J. Math. 26 (1982), 4–9. (1982) MR0638549
  25. Pomerance C., Selfridge J. L., Wagstaff S. S., The pseudoprimes to 25 · 10 9 , , Math. Comp. 35 (1980), 1009–1026. (1980) MR0572872
  26. Ribenboim P., The New Book of Prime Number Records, , Springer, New York, 1996. (1996) Zbl0856.11001MR1377060
  27. Riesel H., Prime Numbers and Computer Methods for Factorization, , Birkhäuser, Boston-Basel-Berlin, 1994. (1994) Zbl0821.11001MR1292250
  28. Rotkiewicz A., 10.1007/BF02843874, , Rend. Circ. Mat. Palermo (2) 11 (1962), 280–282. (1962) Zbl0119.03902MR0166137DOI10.1007/BF02843874
  29. Rotkiewicz A., Sur les nombres pseudopremiers de la forme a x + b , , C.R. Acad. Sci. Paris 257 (1963), 2601–2604. (1963) Zbl0116.03501MR0162757
  30. Rotkiewicz A., Sierpiński W., Sur l’équation diophantienne 2 x - x y = 2 , , Publ. Inst. Math. (Beograd) (N.S.) 4 (18) (1964), 135–137. (1964) MR0171745
  31. Rotkiewicz A., Schinzel A., Sur les nombres pseudopremiers de la forme a x 2 + b x y + c y 2 , , ibidem 258 (1964), 3617–3620. (1964) MR0161828
  32. Rotkiewicz A., Sur les formules donnant des nombres pseudopremiers, , Colloq. Math. 12 (1964), 69–72. (1964) Zbl0129.02703MR0166138
  33. Rotkiewicz A., Pseudoprime Numbers and Their Generalizations, , Stud. Assoc. Fac. Sci. Univ. Novi Sad, 1972, pp. i+169. (1972) Zbl0324.10007MR0330034
  34. Rotkiewicz A., 10.1007/BF02849586, , Rend. Circ. Mat. Palermo (2) 28 (1979), 62–64. (1979) Zbl0425.10009MR0564551DOI10.1007/BF02849586
  35. Rotkiewicz A., van der Poorten A. I., 10.1017/S1446788700021315, , J. Austral. Math. Soc. Ser. A 29 (1980), 316–321. (1980) Zbl0428.10001MR0569519DOI10.1017/S1446788700021315
  36. Sarrus F., Démonstration de la fausseté du théorème énoncé à la page 320 du I X e volume de ce recueil, , Annales de Math. Pure Appl. 10 (1819–20), 184–187. MR1556023
  37. Schinzel A., On primitive prime factors of a n - b n , , Proc. Cambridge Philos. Soc. 58(1962), 555-562. (1962) MR0143728
  38. Sierpiński W., Remarque sur une hypothèse des Chinois concernant les nombres ( 2 n - 2 ) / n , , Colloq. Math. 1 (1948), 9. (1948) MR0023256
  39. Sierpiński W., A selection of Problems in the Theory of Numbers, , Pergamon Press. New York, 1964. (1964) MR0170843
  40. Sierpiński W., Elementary Theory of numbers, , Engl. ed. revised and enlargend by A. Schinzel, Państwowe Wydawnictwo Naukowe, Warszawa, 1988. (1988) MR0930670
  41. Steuerwald R., Über die Kongruenz 2 n - 1 ≡ 1 ( mod n ) , , S.-B. Math.-Nat. Kl., Bayer. Akad. Win., 1947, 177. (1947) MR0030541
  42. Stevenhagen P., 10.1016/1385-7258(87)90009-6, , Nederl. Akad. Wetensch. Indag. Math. 49 (1987), 451–468. (1987) Zbl0635.10010MR0922449DOI10.1016/1385-7258(87)90009-6
  43. Szymiczek K., Note on Fermat numbers, , Elem. Math. 21 (1966), 598. (1966) Zbl0142.28904MR0193056
  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. (1998) Zbl1155.11363MR1632793
  45. Zsigmondy K., 10.1007/BF01692444, , Monastsh. Math. 3 (1892), 265–284. MR1546236DOI10.1007/BF01692444

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