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Welche besonderen Arten von Primzahlen wurden untersucht?

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Die Welt der Primzahlen

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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

  • 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 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 1953 Goldberg, K. A table of Wilson quotients and the third Wilson prime. J. London Math. Soc. 28 (1953), 252–256.

    Article  MATH  MathSciNet  Google Scholar 

  • 1954 Ward, M. Prime divisors of second order recurring sequences. Duke Math. J. 21 (1954), 607–614.

    Article  MATH  MathSciNet  Google Scholar 

  • 1956 Obláth, R. Une propriété des puissances parfaites. Mathesis 65 (1956), 356–364.

    MATH  MathSciNet  Google Scholar 

  • 1956 Riesel, H. Några stora primtal. Elementa 39 (1956), 258–260.

    Google Scholar 

  • 1958 Jarden, D. Recurring Sequences. Riveon Lematematika, Jerusalem 1958 (3. Auflage bei Fibonacci Assoc., San Jose, CA 1973).

    Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 1960 Sierpiński, W. Sur un problème concernant les nombres k · 2n + 1. Elem. Math. 15 (1960), 73–74.

    MATH  MathSciNet  Google Scholar 

  • 1964 Graham, R.L. A Fibonacci-like sequence of composite numbers. Math. Mag. 37 (1964), 322–324.

    Article  MATH  Google Scholar 

  • 1964 Riesel, H. Note on the congruence \(a^{p - 1} \equiv 1\,\left( {{\textrm{mod}} p^2 } \right)\). Math. Comp. 18 (1964), 149–150.

    MATH  MathSciNet  Google Scholar 

  • 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 

  • 1965 Kloss, K.E. Some number theoretic calculations. J. Res. Nat. Bureau of Stand. B, 69 (1965), 335–336.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 1969 Riesel, H. Lucasian criteria for the primality of \(N = h\, \cdot 2^n - 1\). Math. Comp. 23 (1969), 869–875.

    MATH  MathSciNet  Google Scholar 

  • 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 

  • 1975 Johnson, W. Irregular primes and cyclotomic invariants. Math. Comp. 29 (1975), 113–120.

    Article  MATH  MathSciNet  Google Scholar 

  • 1976 Hooley, C. Application of Sieve Methods to the Theory of Numbers. Cambridge Univ. Press, Cambridge 1976.

    Google Scholar 

  • 1978 Wagstaff Jr., S.S. The irregular primes to 125000. Math. Comp. 32 (1978), 583–591.

    MATH  MathSciNet  Google Scholar 

  • 1978 Williams, H.C. Some primes with interesting digit patterns. Math. Comp. 32 (1978), 1306–1310.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 1979 Ribenboim, P. 13 Lectures on Fermat's Last Theorem. Springer-Verlag, New York 1979.

    MATH  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 1980 Powell, B. Primitive densities of certain sets of primes. J. Number Theory 12 (1980), 210–217.

    Article  MATH  MathSciNet  Google Scholar 

  • 1981 Lehmer, D.H. On Fermat’s quotient, base two. Math. Comp. 36 (1981), 289–290.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 1982 Yates, S. Repunits and Repetends. Star Publ. Co., Boynton Beach, FL 1982.

    MATH  Google Scholar 

  • 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 

  • 1983 Keller, W. Factors of Fermat numbers and large primes of the form \(k \cdot 2^n + 1\). Math. Comp. 41 (1983), 661–673.

    MATH  MathSciNet  Google Scholar 

  • 1983 Ribenboim, P. 1093. Math. Intelligencer 5, No. 2 (1983), 28–34.

    Google Scholar 

  • 1985 Dubner, H. Generalized Fermat primes. J. Recr. Math. 18 (1985/86), 279–280.

    Google Scholar 

  • 1985 Lagarias, J.C. The set of primes dividing the Lucas numbers has density \(\frac{2}{3}\) . Pacific J. Math. 118 (1985), 19–23.

    MathSciNet  Google Scholar 

  • 1986 Tzanakis, N. Solution to problem E2956. Amer. Math. Monthly 93 (1986), S. 569.

    MathSciNet  Google Scholar 

  • 1986 Williams, H.C. & Dubner, H. The primality of R1031. Math. Comp. 47 (1986), 703–711.

    MATH  MathSciNet  Google Scholar 

  • 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 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 1989 Dubner, H. Generalized Cullen numbers. J. Recr. Math. 21 (1989), 190–194.

    Google Scholar 

  • 1989 Löh, G. Long chains of nearly doubled primes. Math. Comp. 53 (1989), 751–759.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    MATH  MathSciNet  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 1990 Knuth, D.E. A Fibonacci-like sequence of composite numbers. Math. Mag. 63 (1990), 21–25.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 1991 Fee, G. & Granville, A. The prime factors of Wendt’s binomial circulant determinant. Math. Comp. 57 (1991), 839–848.

    MATH  MathSciNet  Google Scholar 

  • 1991 Keller, W. Woher kommen die größten derzeit bekannten Primzahlen? Mitt. Math. Ges. Hamburg 12 (1991), 211–229.

    MATH  MathSciNet  Google Scholar 

  • 1992 Buhler, J.P., Crandall, R.E. & Sompolski, R.W. Irregular primes to one million. Math. Comp. 59 (1992), 717–722.

    Article  MATH  MathSciNet  Google Scholar 

  • 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.

    Article  MATH  MathSciNet  Google Scholar 

  • 1993 Dubner, H. Generalized repunit primes. Math. Comp. 61 (1993), 927–930.

    Article  MATH  MathSciNet  Google Scholar 

  • 1993 Montgomery, P.L. New solutions of \(a^{p - 1} \equiv 1\,\left( {{\textrm{mod}} p^2 } \right)\). Math. Comp. 61 (1993), 361–363.

    MATH  MathSciNet  Google Scholar 

  • 1994 Crandall, R.E. & Fagin, B. Discrete weighted transforms and large-integer arithmetic. Math. Comp. 62 (1994), 305–324.

    Article  MATH  MathSciNet  Google Scholar 

  • 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 

  • 1994 Suzuki, J. On the generalized Wieferich criteria. Proc. Japan Acad. Sci. A (Math. Sci.), 70 (1994), 230–234.

    Article  MATH  Google Scholar 

  • 1995 Keller, W. New Cullen primes. Math. Comp. 64 (1995), 1733–1741.

    Article  MATH  MathSciNet  Google Scholar 

  • 1995 Keller, W. & Niebuhr, W. Supplement to “New Cullen primes”. Math. Comp. 64 (1995), S39–S46.

    Article  MathSciNet  Google Scholar 

  • 1997 Crandall, R., Dilcher, K. & Pomerance, C. A search for Wieferich and Wilson primes. Math. Comp. 66 (1997), 433–449.

    Article  MATH  MathSciNet  Google Scholar 

  • 1997 Ernvall, R. & Metsänkylä, T. On the p-divisibility of Fermat quotients. Math. Comp. 66 (1997), 1353–1365.

    Article  MATH  MathSciNet  Google Scholar 

  • 1999 Dubner, H. & Keller, W. New Fibonacci and Lucas primes. Math. Comp. 68 (1999), 417–427 and S1–S12.

    Article  MATH  MathSciNet  Google Scholar 

  • 1999 Forbes, T. Prime clusters and Cunningham chains. Math. Comp. 68 (1999), 1739–1747.

    Article  MATH  MathSciNet  Google Scholar 

  • 1999 Ribenboim, P. Fermat's Last Theorem for Amateurs. Springer- Verlag, New York 1999.

    MATH  Google Scholar 

  • 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 

  • 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.

    Article  MATH  Google Scholar 

  • 2002 Dubner, H. Repunit R49081 is a probable prime. Math. Comp. 71 (2002), 833–835.

    Article  MATH  MathSciNet  Google Scholar 

  • 2002 Dubner, H. & Gallot, Y. Distribution of generalized Fermat prime numbers. Math. Comp. 71 (2002), 825–832.

    Article  MATH  MathSciNet  Google Scholar 

  • 2002 Izotov, A.S. Second-order linear recurrences of composite numbers. Fibonacci Quart. 40 (2002), 266–268.

    MATH  MathSciNet  Google Scholar 

  • 2002 Sellers, J.A. & Williams, H.C. On the infinitude of composite NSW numbers. Fibonacci Quart. 40 (2002), 253–254.

    MATH  MathSciNet  Google Scholar 

  • 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 

  • 2005 Keller, W. & Richstein, J. Solutions of the congruence a p−1 ≡ 1 (mod p r). Math. Comp. 74 (2005), 927–936.

    Article  MATH  MathSciNet  Google Scholar 

  • 2005 Knauer, J. & Richstein, J. The continuing search for Wieferich primes. Math. Comp. 74 (2005), 1559–1563.

    Article  MATH  MathSciNet  Google Scholar 

  • 2008 Dorais, F.G. & Klyve, D.W. Near Wieferich primes up to 6.7 × 1015. Unveröffentlichtes Manuskript.

    Google Scholar 

  • 2009 Mossinghoff, M.J. Wieferich pairs and Barker sequences. Designs Codes Cryptogr. 53 (2009), 149–163.

    Article  MATH  MathSciNet  Google Scholar 

  • Slatkevičius, R. & Blazek, J. PrimeGrid. Umfassendes Projekt zur Primzahlsuche in verschiedenen Teilbereichen. http://www.primegrid.com/

  • Caldwell, C. Die größten bekannten Sophie-Germain-Primzahlen. http://primes.utm.edu/largest.html#Sophie

  • Andersen, J.K. Cunningham-Ketten. http://users.cybercity.dk/~dsl522332/math/CunninghamChainrecords.htm

  • Keller, W. & Richstein, J. Fermat-Quotienten qp(a), die durch p teilbar sind. http://www1.uni-hamburg.de/RRZ/W.Keller/FermatQuotient.html

  • Koide, Y. Faktorisierung von Repunit-Zahlen. http://www.h4.dion.ne.jp/~rep/

  • Di Maria, G. Suche nach Repunit-Quasiprimzahlen. http://www.gruppoeratostene.com/ric-repunit/repunit.htm

  • Keller, W. Status des Sierpiński-Problems. http://www.prothsearch.net/sierp.html

  • Keller, W. Status des Riesel-Problems. http://www.prothsearch.net/rieselprob.html

  • Caldwell, C. Die größten bekannten Nicht-Mersenne-Primzahlen. http://primes.utm.edu/primes/lists/short.pdf

  • Leyland, P. Faktorisierung von Cullen- und Woodall-Zahlen. http://www.leyland.vispa.com/numth/factorization/cullenwoodall/cw.htm

  • Löh, G. Verallgemeinerte Cullen-Primzahlen. http://www1.uni-hamburg.de/RRZ/G.Loeh/gc/status.html

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  1. Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    Prof. Dr. Paulo Ribenboim

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Ribenboim, P. (2011). Welche besonderen Arten von Primzahlen wurden untersucht?. In: Die Welt der Primzahlen. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18079-8_5

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