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Welche besonderen Arten von Primzahlen wurden untersucht?

  • Paulo Ribenboim
Chapter
Part of the Springer-Lehrbuch book series (SLB)

Zusammenfassung

Wir waren bereits verschiedenen Arten besonderer Primzahlen begegnet. Zum Beispiel solchen, die Fermat- oder Mersenne-Zahlen sind (siehe Kapitel 2). Ich werde nun weitere Primzahl-Familien besprechen, darunter die regulären Primzahlen, Sophie-Germain-Primzahlen, Wieferich-Primzahlen, Wilson-Primzahlen, Repunit-Primzahlen sowie Primzahlen in linear rekurrenten Folgen zweiter Ordnung.

Reguläre Primzahlen, Sophie-Germain- und Wieferich-Primzahlen entstammen direkt aus Beweisversuchen von Fermats letztem Satz.

Der interessierte Leser möchte dazu vielleicht mein Buch 13 Lectures on Fermat's Last Theorem konsultieren, in dem diese Angelegenheiten genauer besprochen werden. Insbesondere befindet sich darin ein umfassendes Literaturverzeichnis mit zahlreichen klassischen Arbeiten, die im vorliegenden Buch nicht aufgelistet sind.

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Literatur

  1. 1948 Gunderson, N.G. Derivation of Criteria for the First Case of Fermat's Last Theorem and the Combination of these Criteria to Produce a New Lower Bound for the Exponent. Dissertation, Cornell University, 1948, 111 Seiten.Google Scholar
  2. 1951 Dénes, P. An extension of Legendre’s criterion in connection with the first case of Fermat’s last theorem. Publ. Math. Debrecen 2 (1951), 115–120.zbMATHMathSciNetGoogle Scholar
  3. 1953 Goldberg, K. A table of Wilson quotients and the third Wilson prime. J. London Math. Soc. 28 (1953), 252–256.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 1954 Ward, M. Prime divisors of second order recurring sequences. Duke Math. J. 21 (1954), 607–614.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 1956 Obláth, R. Une propriété des puissances parfaites. Mathesis 65 (1956), 356–364.zbMATHMathSciNetGoogle Scholar
  6. 1956 Riesel, H. Några stora primtal. Elementa 39 (1956), 258–260.Google Scholar
  7. 1958 Jarden, D. Recurring Sequences. Riveon Lematematika, Jerusalem 1958 (3. Auflage bei Fibonacci Assoc., San Jose, CA 1973).Google Scholar
  8. 1958 Robinson, R.M. A report on primes of the form k · 2n +1 and on factors of Fermat numbers. Proc. Amer. Math. Soc. 9 (1958), 673–681.zbMATHMathSciNetGoogle Scholar
  9. 1960 Sierpiński, W. Sur un problème concernant les nombres k · 2n + 1. Elem. Math. 15 (1960), 73–74.zbMATHMathSciNetGoogle Scholar
  10. 1964 Graham, R.L. A Fibonacci-like sequence of composite numbers. Math. Mag. 37 (1964), 322–324.zbMATHCrossRefGoogle Scholar
  11. 1964 Riesel, H. Note on the congruence \(a^{p - 1} \equiv 1\,\left( {{\textrm{mod}} p^2 } \right)\). Math. Comp. 18 (1964), 149–150.zbMATHMathSciNetGoogle Scholar
  12. 1964 Siegel, C.L. Zu zwei Bemerkungen Kummers. Nachr. Akad. d. Wiss. Göttingen, Math. Phys. Kl., II, 1964, 51–62. Nachdruck in Gesammelte Abhandlungen (Hrsg. K. Chandrasekharan und H. Maas), Bd. III, 436–442. Springer-Verlag, Berlin 1966.Google Scholar
  13. 1965 Kloss, K.E. Some number theoretic calculations. J. Res. Nat. Bureau of Stand. B, 69 (1965), 335–336.zbMATHMathSciNetGoogle Scholar
  14. 1966 Hasse, H. Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl \(a \ne 0\) von gerader bzw. ungerader Ordnung mod p ist. Math. Ann. 166 (1966), 19–23.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 1966 Kruyswijk, D. On the congruence \(u^{p - 1} \equiv 1\,\left( {{\textrm{mod}} p^2 } \right)\) (niederländisch). Math. Centrum Amsterdam, 1966, 7 Seiten.Google Scholar
  16. 1969 Riesel, H. Lucasian criteria for the primality of \(N = h\, \cdot 2^n - 1\). Math. Comp. 23 (1969), 869–875.zbMATHMathSciNetGoogle Scholar
  17. 1971 Brillhart, J., Tonascia, J. & Weinberger, P.J. On the Fermat quotient. In Computers in Number Theory (Hrsg. A.L. Atkin und B.J. Birch), 213–222. Academic Press, New York 1971.Google Scholar
  18. 1975 Johnson, W. Irregular primes and cyclotomic invariants. Math. Comp. 29 (1975), 113–120.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 1976 Hooley, C. Application of Sieve Methods to the Theory of Numbers. Cambridge Univ. Press, Cambridge 1976.Google Scholar
  20. 1978 Wagstaff Jr., S.S. The irregular primes to 125000. Math. Comp. 32 (1978), 583–591.zbMATHMathSciNetGoogle Scholar
  21. 1978 Williams, H.C. Some primes with interesting digit patterns. Math. Comp. 32 (1978), 1306–1310.zbMATHCrossRefMathSciNetGoogle Scholar
  22. 1979 Erdös, P. & Odlyzko, A.M. On the density of odd integers of the form \(\left( {p - 1} \right)2^{ - n} \) and related questions. J. Number Theory 11 (1979), 257–263.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 1979 Ribenboim, P. 13 Lectures on Fermat's Last Theorem. Springer-Verlag, New York 1979.zbMATHGoogle Scholar
  24. 1979 Williams, H.C. & Seah, E. Some primes of the form \(\left( {a^n - 1} \right)/\left( {a - 1} \right)\). Math. Comp. 33 (1979), 1337–1342.zbMATHMathSciNetGoogle Scholar
  25. 1980 Newman, M., Shanks, D. & Williams, H.C. Simple groups of square order and an interesting sequence of primes. Acta Arith. 38 (1980), 129–140.zbMATHMathSciNetGoogle Scholar
  26. 1980 Powell, B. Primitive densities of certain sets of primes. J. Number Theory 12 (1980), 210–217.zbMATHCrossRefMathSciNetGoogle Scholar
  27. 1981 Lehmer, D.H. On Fermat’s quotient, base two. Math. Comp. 36 (1981), 289–290.zbMATHMathSciNetGoogle Scholar
  28. 1982 Powell, B. Problem E 2956 (The existence of small prime solutions of \(x^{p - 1} \equiv\!\!\!\!{/} 1\ \ \left( {\textrm{mod}} p^{2}\right)\)). Amer. Math. Monthly 89 (1982), S. 498.CrossRefGoogle Scholar
  29. 1982 Yates, S. Repunits and Repetends. Star Publ. Co., Boynton Beach, FL 1982.zbMATHGoogle Scholar
  30. 1983 Jaeschke, G. On the smallest k such that \(k \cdot 2^N + 1\) are composite. Math. Comp. 40 (1983), 381–384; Corrigendum, 45 (1985), S. 637.Google Scholar
  31. 1983 Keller, W. Factors of Fermat numbers and large primes of the form \(k \cdot 2^n + 1\). Math. Comp. 41 (1983), 661–673.zbMATHMathSciNetGoogle Scholar
  32. 1983 Ribenboim, P. 1093. Math. Intelligencer 5, No. 2 (1983), 28–34.Google Scholar
  33. 1985 Dubner, H. Generalized Fermat primes. J. Recr. Math. 18 (1985/86), 279–280.Google Scholar
  34. 1985 Lagarias, J.C. The set of primes dividing the Lucas numbers has density \(\frac{2}{3}\) . Pacific J. Math. 118 (1985), 19–23.MathSciNetGoogle Scholar
  35. 1986 Tzanakis, N. Solution to problem E2956. Amer. Math. Monthly 93 (1986), S. 569.MathSciNetGoogle Scholar
  36. 1986 Williams, H.C. & Dubner, H. The primality of R1031. Math. Comp. 47 (1986), 703–711.zbMATHMathSciNetGoogle Scholar
  37. 1987 Granville, A. Diophantine Equations with Variable Exponents with Special Reference to Fermat's Last Theorem. Dissertation, Queen’s University, Kingston, Ontario 1987, 207 Seiten.Google Scholar
  38. 1987 Rotkiewicz, A. Note on the diophantine equation \(1 + x + x^2 + \cdot \cdot \cdot + x^n = y^m \). Elem. Math. 42 (1987), S. 76.zbMATHMathSciNetGoogle Scholar
  39. 1988 Brillhart, J., Montgomery, P.L. & Silverman, R.D. Tables of Fibonacci and Lucas factorizations, and Supplement. Math. Comp. 50 (1988), 251–260 und S1–S15.CrossRefMathSciNetGoogle Scholar
  40. 1988 Gonter, R.H. & Kundert, E.G. Wilson's theorem \(\left( {n - 1} \right)! \equiv - 1\,\,\left( {{\textrm{mod}} \,p^2 } \right)\) has been computed up to 10,000,000. Fourth SIAM Conference on Discrete Mathematics, San Francisco 1988.Google Scholar
  41. 1988 Granville, A. & Monagan, M.B. The first case of Fermat’s last theorem is true for all prime exponents up to 714,591,416,091, 389. Trans. Amer. Math. Soc. 306 (1988), 329–359.zbMATHMathSciNetGoogle Scholar
  42. 1989 Dubner, H. Generalized Cullen numbers. J. Recr. Math. 21 (1989), 190–194.Google Scholar
  43. 1989 Löh, G. Long chains of nearly doubled primes. Math. Comp. 53 (1989), 751–759.zbMATHCrossRefMathSciNetGoogle Scholar
  44. 1989 Tanner, J.W. & Wagstaff Jr., S.S. New bound for the first case of Fermat’s last theorem. Math. Comp. 53 (1989), 743–750.zbMATHMathSciNetGoogle Scholar
  45. 1990 Brown, J., Noll, L.C., Parady, B.K., Smith, J.F., Smith, G.W. & Zarantonello, S. Letter to the editor. Amer. Math. Monthly 97 (1990), S. 214.CrossRefGoogle Scholar
  46. 1990 Knuth, D.E. A Fibonacci-like sequence of composite numbers. Math. Mag. 63 (1990), 21–25.zbMATHCrossRefMathSciNetGoogle Scholar
  47. 1991 Aaltonen, M. & Inkeri, K. Catalan’s equation \(x^p - y^q = 1\) and related congruences. Math. Comp. 56 (1991), 359–370. Nachdruck in Collected Papers of Kustaa Inkeri (Hrsg. T. Metsänkylä und P. Ribenboim), Queen’s Papers in Pure and Appl. Math. 91. Queen’s Univ., Kingston, Ontario 1992.Google Scholar
  48. 1991 Fee, G. & Granville, A. The prime factors of Wendt’s binomial circulant determinant. Math. Comp. 57 (1991), 839–848.zbMATHMathSciNetGoogle Scholar
  49. 1991 Keller, W. Woher kommen die größten derzeit bekannten Primzahlen? Mitt. Math. Ges. Hamburg 12 (1991), 211–229.zbMATHMathSciNetGoogle Scholar
  50. 1992 Buhler, J.P., Crandall, R.E. & Sompolski, R.W. Irregular primes to one million. Math. Comp. 59 (1992), 717–722.zbMATHCrossRefMathSciNetGoogle Scholar
  51. 1993 Buhler, J.P., Crandall, R.E., Ernvall, R. & Metsänkylä, T. Irregular primes and cyclotomic invariants to four million. Math. Comp. 61 (1993), 151–153.zbMATHCrossRefMathSciNetGoogle Scholar
  52. 1993 Dubner, H. Generalized repunit primes. Math. Comp. 61 (1993), 927–930.zbMATHCrossRefMathSciNetGoogle Scholar
  53. 1993 Montgomery, P.L. New solutions of \(a^{p - 1} \equiv 1\,\left( {{\textrm{mod}} p^2 } \right)\). Math. Comp. 61 (1993), 361–363.zbMATHMathSciNetGoogle Scholar
  54. 1994 Crandall, R.E. & Fagin, B. Discrete weighted transforms and large-integer arithmetic. Math. Comp. 62 (1994), 305–324.zbMATHCrossRefMathSciNetGoogle Scholar
  55. 1994 Gonter, R.H. & Kundert, E.G. All prime numbers up to 18,876,041 have been tested without ńding a new Wilson prime. Unveröffentlichtes Manuskript, Amherst, MA 1994, 10 Seiten.Google Scholar
  56. 1994 Suzuki, J. On the generalized Wieferich criteria. Proc. Japan Acad. Sci. A (Math. Sci.), 70 (1994), 230–234.zbMATHCrossRefGoogle Scholar
  57. 1995 Keller, W. New Cullen primes. Math. Comp. 64 (1995), 1733–1741.zbMATHCrossRefMathSciNetGoogle Scholar
  58. 1995 Keller, W. & Niebuhr, W. Supplement to “New Cullen primes”. Math. Comp. 64 (1995), S39–S46.CrossRefMathSciNetGoogle Scholar
  59. 1997 Crandall, R., Dilcher, K. & Pomerance, C. A search for Wieferich and Wilson primes. Math. Comp. 66 (1997), 433–449.zbMATHCrossRefMathSciNetGoogle Scholar
  60. 1997 Ernvall, R. & Metsänkylä, T. On the p-divisibility of Fermat quotients. Math. Comp. 66 (1997), 1353–1365.zbMATHCrossRefMathSciNetGoogle Scholar
  61. 1999 Dubner, H. & Keller, W. New Fibonacci and Lucas primes. Math. Comp. 68 (1999), 417–427 and S1–S12.zbMATHCrossRefMathSciNetGoogle Scholar
  62. 1999 Forbes, T. Prime clusters and Cunningham chains. Math. Comp. 68 (1999), 1739–1747.zbMATHCrossRefMathSciNetGoogle Scholar
  63. 1999 Ribenboim, P. Fermat's Last Theorem for Amateurs. Springer- Verlag, New York 1999.zbMATHGoogle Scholar
  64. 2000 Pinch, R.G.E. The pseudoprimes up to 1013. In Proc. Fourth Int. Symp. on Algorithmic Number Th. (Hrsg. W. Bosma). Lecture Notes in Computer Sci. #1838, 459–474. Springer-Verlag, New York 2000.Google Scholar
  65. 2001 Buhler, J., Crandall, R., Ernvall, R., Metsänkylä, T. & Shokrollahi, M.A. Irregular primes and cyclotomic invariants to 12 million. J. Symbolic Comp. 31 (2001), 89–96.zbMATHCrossRefGoogle Scholar
  66. 2002 Dubner, H. Repunit R49081 is a probable prime. Math. Comp. 71 (2002), 833–835.zbMATHCrossRefMathSciNetGoogle Scholar
  67. 2002 Dubner, H. & Gallot, Y. Distribution of generalized Fermat prime numbers. Math. Comp. 71 (2002), 825–832.zbMATHCrossRefMathSciNetGoogle Scholar
  68. 2002 Izotov, A.S. Second-order linear recurrences of composite numbers. Fibonacci Quart. 40 (2002), 266–268.zbMATHMathSciNetGoogle Scholar
  69. 2002 Sellers, J.A. & Williams, H.C. On the infinitude of composite NSW numbers. Fibonacci Quart. 40 (2002), 253–254.zbMATHMathSciNetGoogle Scholar
  70. 2004 Vsemirnov, M. A new Fibonacci-like sequence of composite numbers. J. Integer Seq. 7 (2004), Art. 04.3.7, 1–3 (elektronisch).Google Scholar
  71. 2005 Keller, W. & Richstein, J. Solutions of the congruence a p−1 1 (mod p r). Math. Comp. 74 (2005), 927–936.zbMATHCrossRefMathSciNetGoogle Scholar
  72. 2005 Knauer, J. & Richstein, J. The continuing search for Wieferich primes. Math. Comp. 74 (2005), 1559–1563.zbMATHCrossRefMathSciNetGoogle Scholar
  73. 2008 Dorais, F.G. & Klyve, D.W. Near Wieferich primes up to 6.7 × 1015. Unveröffentlichtes Manuskript.Google Scholar
  74. 2009 Mossinghoff, M.J. Wieferich pairs and Barker sequences. Designs Codes Cryptogr. 53 (2009), 149–163.zbMATHCrossRefMathSciNetGoogle Scholar
  75. Slatkevičius, R. & Blazek, J. PrimeGrid. Umfassendes Projekt zur Primzahlsuche in verschiedenen Teilbereichen. http://www.primegrid.com/
  76. Caldwell, C. Die größten bekannten Sophie-Germain-Primzahlen. http://primes.utm.edu/largest.html#Sophie
  77. Keller, W. & Richstein, J. Fermat-Quotienten qp(a), die durch p teilbar sind. http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html
  78. Koide, Y. Faktorisierung von Repunit-Zahlen. http://www.h4.dion.ne.jp/~rep/
  79. Di Maria, G. Suche nach Repunit-Quasiprimzahlen. http://www.gruppoeratostene.com/ric-repunit/repunit.htm
  80. Keller, W. Status des Sierpiński-Problems. http://www.prothsearch.net/sierp.html
  81. Keller, W. Status des Riesel-Problems. http://www.prothsearch.net/rieselprob.html
  82. Caldwell, C. Die größten bekannten Nicht-Mersenne-Primzahlen. http://primes.utm.edu/primes/lists/short.pdf
  83. Leyland, P. Faktorisierung von Cullen- und Woodall-Zahlen. http://www.leyland.vispa.com/numth/factorization/cullenwoodall/cw.htm
  84. Löh, G. Verallgemeinerte Cullen-Primzahlen. http://www1.uni-hamburg.de/RRZ/G.Loeh/gc/status.html

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Paulo Ribenboim
    • 1
  1. 1.Department of Mathematics and StatisticsQueen’s UniversityKingstonCanada

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