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Wie kann man Primzahlen erkennen?

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Die Welt der Primzahlen

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Zusammenfassung

Dass die Aufgabe, die Primzahlen von den zusammengesetzten zu unterscheiden und letztere in ihre Primfactoren zu zerlegen, zu den wichtigsten und nützlichsten der gesamten Arithmetik gehört und die Bemühungen und den Scharfsinn sowohl der alten wie auch der neueren Geometer in Anspruch genommen hat, ist so bekannt, dass es überflüssig wäre, hierüber viele Worte zu verlieren. [ … ] Ausserdem aber dürfte es die Würde der Wissenschaft erheischen, alle Hülfsmittel zur Lösung jenes so eleganten und berühmten Problems fleissig zu vervollkommnen.

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Literatur

  • 1801 Gauß, C.F. Disquisitiones Arithmeticae. G. Fleischer, Leipzig 1801. Deutsche Ausgabe von H. Maser, Berlin 1889; Nachdruck bei Chelsea, Bronx, NY 1965. Englische Übersetzung von A.A. Clarke bei Yale Univ. Press, New Haven 1966; überarbeitung von W.C. Waterhouse beim Springer-Verlag, New York 1986.

    Google Scholar 

  • 1844 Eisenstein, F.G. Aufgaben. J. Reine Angew. Math. 27 (1844), S. 87. Nachdruck in Mathematische Werke, Bd. I, S. 112. Chelsea, Bronx, NY 1975.

    Google Scholar 

  • 1852 Kummer, E.E. Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen. J. Reine Angew. Math. 44 (1852), 93–146. Nachdruck in Collected Papers (Hrsg. A. Weil), Bd. I, 485–538. Springer-Verlag, New York 1975.

    Google Scholar 

  • 1876 Lucas, E. Sur la recherche des grands nombres premiers. Assoc. Française p. l'Avanc. des Sciences 5 (1876), 61–68.

    Google Scholar 

  • 1877 Pepin, T. Sur la formule \({2^{{2^n}}} + 1\). C.R. Acad. Sci. Paris 85 (1877), 329–331.

    Google Scholar 

  • 1878 Lucas, E. Théorie des fonctions numériques simplement périodiques. Amer. J. Math. 1 (1878), 184–240 und 289–321.

    MathSciNet  Google Scholar 

  • 1878 Proth, F. Théorèmes sur les nombres premiers. C.R. Acad. Sci. Paris 85 (1877), 329–331.

    Google Scholar 

  • 1886 Bang, A.S. Taltheoretiske Undersøgelser. Tidskrift f. Math. (5) 4 (1886), 70–80 und 130–137.

    Google Scholar 

  • 1891 Lucas, E. Théorie des Nombres. Gauthier-Villars, Paris 1891. Nachdruck bei A. Blanchard, Paris 1961.

    MATH  Google Scholar 

  • 1892 Zsigmondy, K. Zur Theorie der Potenzreste. Monatsh. f. Math. Phys. 3 (1892), 265–284.

    MathSciNet  Google Scholar 

  • 1899 Korselt, A. Problème chinois. L'Interm. des Math. 6 (1899), 142–143.

    Google Scholar 

  • 1903 Malo, E. Nombres qui, sans être premiers, vérifient exceptionnellement une congruence de Fermat. L'Interm. des Math. 10 (1903), S. 88.

    Google Scholar 

  • 1904 Birkhoff, G.D. & Vandiver, H.S. On the integral divisors of a n − b n. Annals of Math. (2) 5 (1904), 173–180.

    Google Scholar 

  • 1904 Cipolla, M. Sui numeri composti P, che verificano la congruenza di Fermat a P−1 ≡ 1 (mod P). Annali di Matematica (3) 9 (1904), 139–160.

    Google Scholar 

  • 1912 Carmichael, R.D. On composite numbers P which satisfy the Fermat congruence a P−1 ≡ 1 (mod P). Amer. Math. Monthly 19 (1912), 22–27.

    MathSciNet  Google Scholar 

  • 1913 Carmichael, R.D. On the numerical factors of the arithmetic forms α n ± β n. Annals of Math. (2) 15 (1913), 30–70.

    Google Scholar 

  • 1913 Dickson, L.E. Finiteness of odd perfect and primitive abundant numbers with n distinct prime factors. Amer. J. Math. 35 (1913), 413–422. Nachdruck in The Collected Mathematical Papers (Hrsg. A.A. Albert), Bd. I, 349–358. Chelsea, Bronx, NY 1975.

    Google Scholar 

  • 1914 Pocklington, H.C. The determination of the prime or composite nature of large numbers by Fermat’s theorem. Proc. Cambridge Phil. Soc. 18 (1914/16), 29–30.

    MATH  Google Scholar 

  • 1921 Pirandello, L. Il Fu Mattia Pascal. Bemporad & Figlio, Firenze 1921. Deutsche Übersetzung bei Wagenbach, Berlin 2008.

    Google Scholar 

  • 1922 Carmichael, R.D. Note on Euler’s φ-function. Bull. Amer. Math. Soc. 28 (1922), 109–110.

    MATH  MathSciNet  Google Scholar 

  • 1925 Cunningham, A.J.C. & Woodall, H.J. Factorization of y n± 1, y = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n). Hodgson, London 1925.

    Google Scholar 

  • 1929 Pillai, S.S. On some functions connected with φ(n). Bull. Amer. Math. Soc. 35 (1929), 832–836.

    MATH  MathSciNet  Google Scholar 

  • 1930 Lehmer, D.H. An extended theory of Lucas’ functions. Annals of Math. 31 (1930), 419–448. Nachdruck in Selected Papers (Hrsg. D. McCarthy), Bd. I, 11–48. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1932 Lehmer, D.H. On Euler’s totient function. Bull. Amer. Math. Soc. 38 (1932), 745–751. Nachdruck in Selected Papers (Hrsg. D. Mc- Carthy), Bd. I, 319–325. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1932 Western, A.E. On Lucas’ and Pepin’s tests for the primeness of Mersenne’s numbers. J. London Math. Soc. 7 (1932), 130–137.

    MATH  Google Scholar 

  • 1935 Archibald, R.C. Mersenne’s numbers. Scripta Math. 3 (1935), 112–119.

    MATH  Google Scholar 

  • 1935 Lehmer, D.H. On Lucas’ test for the primality of Mersenne numbers. J. London Math. Soc. 10 (1935), 162–165. Nachdruck in Selected Papers (Hrsg. D. McCarthy), Bd. I, 86–89. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1936 Lehmer, D.H. On the converse of Fermat’s theorem. Amer. Math. Monthly 43 (1936), 347–354. Nachdruck in Selected Papers (Hrsg. D. McCarthy), Bd. I, 90–95. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1939 Chernick, J. On Fermat’s simple theorem. Bull. Amer. Math. Soc. 45 (1939), 269–274.

    MathSciNet  Google Scholar 

  • 1944 Pillai, S.S. On the smallest primitive root of a prime. J. Indian Math. Soc. (N.S.) 8 (1944), 14–17.

    MATH  MathSciNet  Google Scholar 

  • 1944 Schuh, F. Can n−1 be divisible by φ(n) when n is composite? (niederländisch). Mathematica, Zutphen (B) 12 (1944), 102–107.

    Google Scholar 

  • 1945 Kaplansky, I. Lucas’ tests for Mersenne numbers. Amer. Math. Monthly 52 (1945), 188–190.

    MATH  MathSciNet  Google Scholar 

  • 1946 Erdös, P. Problem 4221 (To solve the equation φ(n) = k! for every k ≥ 1). Amer. Math. Monthly 53 (1946), S. 537.

    MathSciNet  Google Scholar 

  • 1947 Klee, V.L. On a conjecture of Carmichael. Bull. Amer. Math. Soc. 53 (1947), 1183–1186.

    MATH  MathSciNet  Google Scholar 

  • 1947 Lehmer, D.H. On the factors of 2n ± 1. Bull. Amer. Math. Soc. 53 (1947), 164–167. Nachdruck in Selected Papers (Hrsg. D. McCarthy), Bd. III, 1081–1084. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1948 Lambek, J. Solution of problem 4221 (proposed by P. Erdös). Amer. Math. Monthly 55 (1948), S. 103.

    MathSciNet  Google Scholar 

  • 1948 Ore, O. On the averages of the divisors of a number. Amer. Math. Monthly 55 (1948), 615–619.

    MATH  MathSciNet  Google Scholar 

  • 1948 Steuerwald, R. Über die Kongruenz a n−1 ≡ 1 (mod n). Sitzungsber. math.-naturw. Kl. Bayer. Akad. Wiss. München, 1948, 69–70.

    Google Scholar 

  • 1949 Erdös, P. On the converse of Fermat’s theorem. Amer. Math. Monthly 56 (1949), 623–624.

    MATH  MathSciNet  Google Scholar 

  • 1949 Fridlender, V.R. On the least nth power non-residue (russisch). Doklady Akad. Nauk SSSR (N.S.) 66 (1949), 351–352.

    MATH  MathSciNet  Google Scholar 

  • 1949 Shapiro, H.N. Note on a theorem of Dickson. Bull. Amer. Math. Soc. 55 (1949), 450–452.

    MATH  MathSciNet  Google Scholar 

  • 1950 Beeger, N.G.W.H. On composite numbers n for which a n−1 ≡ 1 (mod n), for every a, prime to n. Scripta Math. 16 (1950), 133–135.

    MATH  MathSciNet  Google Scholar 

  • 1950 Giuga, G. Su una presumibile proprietà caratteristica dei numeri primi. Ist. Lombardo Sci. Lett. Rend. Cl. Sci. Mat. Nat. (3) 14 (83), (1950), 511–528.

    Google Scholar 

  • 1950 Gupta, H. On a problem of Erdös. Amer. Math. Monthly 57 (1950), 326–329.

    MATH  MathSciNet  Google Scholar 

  • 1950 Kanold, H.-J. Sätze über Kreisteilungspolynome und ihre Anwendungen auf einige zahlentheoretische Probleme, II. J. Reine Angew. Math. 187 (1950), 355–366.

    MathSciNet  Google Scholar 

  • 1950 Salié, H. Über den kleinsten positiven quadratischen Nichtrest einer Primzahl. Math. Nachr. 3 (1949), 7–8.

    MathSciNet  Google Scholar 

  • 1950 Somayajulu, B.S.K.R. On Euler’s totient function φ(n). Math. Student 18 (1950), 31–32.

    MathSciNet  Google Scholar 

  • 1951 Beeger, N.G.W.H. On even numbers m dividing 2m −2. Amer. Math. Monthly 58 (1951), 553–555.

    MATH  MathSciNet  Google Scholar 

  • 1951 Morrow, D.C. Some properties of D numbers. Amer. Math. Monthly 58 (1951), 329–330.

    MATH  MathSciNet  Google Scholar 

  • 1952 Duparc, H.J.A. On Carmichael numbers. Simon Stevin 29 (1952), 21–24.

    MATH  MathSciNet  Google Scholar 

  • 1952 Grün, O. Über ungerade vollkommene Zahlen. Math. Zeitschrift 55 (1952), 353–354.

    MATH  Google Scholar 

  • 1953 Knödel, W. Carmichaelsche Zahlen. Math. Nachr. 9 (1953), 343–350.

    MATH  MathSciNet  Google Scholar 

  • 1953 Selfridge, J.L. Factors of Fermat numbers. Math. Comp. 7 (1953), 274–275.

    Google Scholar 

  • 1953 Touchard, J. On prime numbers and perfect numbers. Scripta Math. 19 (1953), 35–39.

    MATH  MathSciNet  Google Scholar 

  • 1954 Garcia, M. On numbers with integral harmonic mean. Amer. Math. Monthly 61 (1954), 89–96.

    MATH  MathSciNet  Google Scholar 

  • 1954 Kanold, H.-J. Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen. Math. Zeitschrift 61 (1954), 180–185.

    MATH  MathSciNet  Google Scholar 

  • 1954 Robinson, R.M. Mersenne and Fermat numbers. Proc. Amer. Math. Soc. 5 (1954), 842–846.

    MATH  MathSciNet  Google Scholar 

  • 1954 Schinzel, A. Quelques théorèmes sur les fonctions φ(n) et σ(n). Bull. Acad. Polon. Sci., Cl. III, 2 (1954), 467–469.

    MATH  MathSciNet  Google Scholar 

  • 1954 Schinzel, A. Generalization of a theorem of B.S.K.R. Somayajulu on the Euler’s function φ(n). Ganita 5 (1954), 123–128.

    MATH  MathSciNet  Google Scholar 

  • 1954 Schinzel, A. & Sierpiński, W. Sur quelques propriétés des fonctions φ(n) et σ(n). Bull. Acad. Polon. Sci., Cl. III, 2 (1954), 463–466.

    MATH  MathSciNet  Google Scholar 

  • 1955 Artin, E. The orders of the linear groups. Comm. Pure and Appl. Math. 8 (1955), 355–365. Nachdruck in Collected Papers (Hrsg. S. Lang und J.T. Tate), 387–397. Addison-Wesley, Reading, MA 1965.

    Google Scholar 

  • 1955 Hornfeck, B. Zur Dichte der Menge der vollkommenen Zahlen. Arch. Math. 6 (1955), 442–443.

    MATH  MathSciNet  Google Scholar 

  • 1955 Laborde, P. A note on the even perfect numbers. Amer. Math. Monthly 62 (1955), 348–349.

    MATH  MathSciNet  Google Scholar 

  • 1956 Hornfeck, B. Bemerkung zu meiner Note über vollkommene Zahlen. Arch. Math. 7 (1956), S. 273.

    Google Scholar 

  • 1956 Kanold, H.-J. Über einen Satz von L.E. Dickson, II. Math. Ann. 131 (1956), 246–255.

    Google Scholar 

  • 1956 Schinzel, A. Sur l’équation φ(x) = m. Elem. Math. 11 (1956), 75–78.

    MATH  MathSciNet  Google Scholar 

  • 1956 Schinzel, A. Sur un problème concernant la fonction φ(n). Czechoslovak Math. J. 6 (81), (1956), 164–165.

    MathSciNet  Google Scholar 

  • 1957 Hornfeck, B. & Wirsing, E. Über die Häufigkeit vollkommener Zahlen. Math. Ann. 133 (1957), 431–438.

    MATH  MathSciNet  Google Scholar 

  • 1957 Kanold, H.-J. Über die Verteilung der vollkommenen Zahlen und allgemeinerer Zahlenmengen. Math. Ann. 132 (1957), 442–450.

    MATH  MathSciNet  Google Scholar 

  • 1958 Erdös, P. Some remarks on Euler’s φ-function. Acta Arith. 4 (1958), 10–19.

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  • 1958 Perisastri, M. A note on odd perfect numbers. Math. Student 26 (1958), 179–181.

    MathSciNet  Google Scholar 

  • 1958 Schinzel, A. Sur les nombres composés n qui divisent a n − a. Rend. Circ. Mat. Palermo (2) 7 (1958), 37–41.

    Google Scholar 

  • 1958 Sierpiński, W. Sur les nombres premiers de la forme n n + 1. L'Enseign. Math. (2) 4 (1958), 211–212.

    Google Scholar 

  • 1959 Rotkiewicz, A. Sur les nombres pairs n pour lesquels les nombres a n b − ab n, respectivement a n−1 − b n−1 sont divisibles par n. Rend. Circ. Mat. Palermo (2) 8 (1959), 341–342.

    Google Scholar 

  • 1959 Satyanarayana, M. Odd perfect numbers. Math. Student 27 (1959), 17–18.

    MathSciNet  Google Scholar 

  • 1959 Schinzel, A. Sur les nombres composés n qui divisent a n − a. Rend. Circ. Mat. Palermo (2) 7 (1958), 1–5.

    Google Scholar 

  • 1959 Wirsing, E. Bemerkung zu der Arbeit über vollkommene Zahlen. Math. Ann. 137 (1959), 316–318.

    MATH  MathSciNet  Google Scholar 

  • 1960 Inkeri, K. Tests for primality. Annales Acad. Sci. Fennicae, Ser. A, I, 279, Helsinki 1960, 19 Seiten. 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 

  • 1961 Ward, M. The prime divisors of Fibonacci numbers. Pacific J. Math. 11 (1961), 379–389.

    MATH  MathSciNet  Google Scholar 

  • 1962 Burgess, D.A. On character sums and L-series. Proc. London Math. Soc. (3) 12 (1962), 193–206.

    Google Scholar 

  • 1962 Crocker, R. A theorem on pseudo-primes. Amer. Math. Monthly 69 (1962), p. 540.

    MATH  MathSciNet  Google Scholar 

  • 1962 Mąkowski, A. Generalization of Morrow’s D numbers. Simon Stevin 36 (1962), S. 71.

    MATH  MathSciNet  Google Scholar 

  • 1962 Schinzel, A. The intrinsic divisors of Lehmer numbers in the case of negative discriminant. Arkiv för Mat. 4 (1962), 413–416.

    MATH  MathSciNet  Google Scholar 

  • 1962 Schinzel, A. On primitive prime factors of a n−b n. Proc. Cambridge Phil. Soc. 58 (1962), 555–562.

    MATH  MathSciNet  Google Scholar 

  • 1962 Shanks, D. Solved and Unsolved Problems in Number Theory. Spartan, Washington 1962 (3. Auflage bei Chelsea, Bronx, NY 1985).

    Google Scholar 

  • 1963 Schinzel, A. On primitive prime factors of Lehmer numbers, I. Acta Arith. 8 (1963), 211–223.

    Google Scholar 

  • 1963 Schinzel, A. On primitive prime factors of Lehmer numbers, II. Acta Arith. 8 (1963), 251–257.

    MathSciNet  Google Scholar 

  • 1963 Suryanarayana, D. On odd perfect numbers, II. Proc. Amer. Math. Soc. 14 (1963), 896–904.

    MATH  MathSciNet  Google Scholar 

  • 1964 Biermann, K.-R. Thomas Clausen, Mathematiker und Astronom. J. Reine Angew. Math. 216 (1964), 159–198.

    MATH  MathSciNet  Google Scholar 

  • 1964 Gillies, D.B. Three new Mersenne primes and a statistical theory. Math. Comp. 18 (1964), 93–98.

    MATH  MathSciNet  Google Scholar 

  • 1964 Lehmer, E. On the infinitude of Fibonacci pseudo-primes. Fibonacci Quart. 2 (1964), 229–230.

    MATH  Google Scholar 

  • 1965 Erdös, P. Some recent advances and current problems in number theory. In Lectures in Modern Mathematics, Bd. III (Hrsg. T.L. Saaty), 196–244. Wiley, New York 1965.

    Google Scholar 

  • 1965 Rotkiewicz, A. Sur les nombres de Mersenne dépourvus de facteurs carrés et sur les nombres naturels n tels que n 2 | 2n − 2. Matem. Vesnik (Beograd) 2 (17), (1965), 78–80.

    MathSciNet  Google Scholar 

  • 1966 Grosswald, E. Topics from the Theory of Numbers. Macmillan, New York 1966 (2. Auflage bei Birkhäuser, Boston 1984).

    MATH  Google Scholar 

  • 1966 Muskat, J.B. On divisors of odd perfect numbers. Math. Comp. 20 (1966), 141–144.

    MATH  MathSciNet  Google Scholar 

  • 1967 Brillhart, J. & Selfridge, J.L. Some factorizations of 2n ± 1 and related results. Math. Comp. 21 (1967), 87–96 und S. 751.

    MATH  MathSciNet  Google Scholar 

  • 1967 Mozzochi, C.J. A simple proof of the Chinese remainder theorem. Amer. Math. Monthly 74 (1967), S. 998.

    MATH  MathSciNet  Google Scholar 

  • 1970 Lieuwens, E. Do there exist composite numbers M for which kφ(M) = M − 1 holds? Nieuw Arch. Wisk. (3) 18 (1970), 165–169.

    Google Scholar 

  • 1970 Parberry, E.A. On primes and pseudo-primes related to the Fibonacci sequence. Fibonacci Quart. 8 (1970), 49–60.

    MATH  MathSciNet  Google Scholar 

  • 1970 Suryanarayana, D. & Hagis Jr., P. A theorem concerning odd perfect numbers. Fibonacci Quart. 8 (1970), 337–346.

    MathSciNet  Google Scholar 

  • 1971 Lieuwens, E. Fermat Pseudo-Primes. Dissertation, Delft 1971.

    Google Scholar 

  • 1971 Morrison, M.A. & Brillhart, J. The factorization of F 7. Bull. Amer. Math. Soc. 77 (1971), S. 264.

    MATH  MathSciNet  Google Scholar 

  • 1971 Schönhage, A. & Strassen, V. Schnelle Multiplikation grosser Zahlen. Computing 7 (1971), 281–292.

    MATH  Google Scholar 

  • 1972 Hagis Jr., P. & McDaniel, W.L. A new result concerning the structure of odd perfect numbers. Proc. Amer. Math. Soc. 32 (1972), 13–15.

    MATH  MathSciNet  Google Scholar 

  • 1972 Mills, W.H. On a conjecture of Ore. Proc. 1972 Number Theory Conf. in Boulder, 142–146.

    Google Scholar 

  • 1972 Ribenboim, P. Algebraic Numbers. Wiley-Interscience, New York 1972 (erweiterte Neuauflage beim Springer-Verlag, New York 2001).

    MATH  Google Scholar 

  • 1972 Rotkiewicz, A. Pseudoprime Numbers and their Generalizations. Stud. Assoc. Fac. Sci. Univ. Novi Sad, 1972.

    Google Scholar 

  • 1973 Grosswald, E. Contributions to the theory of Euler’s function φ(x). Bull. Amer. Math. Soc. 79 (1973), 327–341.

    MathSciNet  Google Scholar 

  • 1973 Hagis Jr., P. A lower bound for the set of odd perfect numbers. Math. Comp. 27 (1973), 951–953.

    MATH  MathSciNet  Google Scholar 

  • 1973 Rotkiewicz, A. On pseudoprimes with respect to the Lucas sequences. Bull. Acad. Polon. Sci. 21 (1973), 793–797.

    MATH  MathSciNet  Google Scholar 

  • 1974 Ligh, S. & Neal, L. A note on Mersenne numbers. Math. Mag. 47 (1974), 231–233.

    MATH  MathSciNet  Google Scholar 

  • 1974 Pomerance, C. On Carmichael’s conjecture. Proc. Amer. Math. Soc. 43 (1974), 297–298.

    MATH  MathSciNet  Google Scholar 

  • 1974 Schinzel, A. Primitive divisors of the expression A n − B n in algebraic number fields. J. Reine Angew. Math. 268/269 (1974), 27–33.

    MathSciNet  Google Scholar 

  • 1974 Sinha, T.N. Note on perfect numbers. Math. Student 42 (1974), S. 336.

    MathSciNet  Google Scholar 

  • 1975 Brillhart, J., Lehmer, D.H. & Selfridge, J.L. New primality criteria and factorizations of 2m ± 1. Math. Comp. 29 (1975), 620–647.

    MATH  MathSciNet  Google Scholar 

  • 1975 Guy, R.K. How to factor a number. Proc. Fifth Manitoba Conf. Numerical Math., 1975, 49–89 (Congressus Numerantium, XVI, Winnipeg, Manitoba 1975).

    Google Scholar 

  • 1975 Hagis Jr., P. & McDaniel, W.L. On the largest prime divisor of an odd perfect number. Math. Comp. 29 (1975), 922–924.

    MATH  MathSciNet  Google Scholar 

  • 1975 Morrison, M.A. A note on primality testing using Lucas sequences. Math. Comp. 29 (1975), 181–182.

    MATH  MathSciNet  Google Scholar 

  • 1975 Pomerance, C. The second largest prime factor of an odd perfect number. Math. Comp. 29 (1975), 914–921.

    MATH  MathSciNet  Google Scholar 

  • 1975 Pratt, V.R. Every prime has a succinct certificate. SIAM J. Comput. 4 (1975), 214–220.

    MATH  MathSciNet  Google Scholar 

  • 1975 Stewart, C.L. The greatest prime factor of a n −b n. Acta Arith. 26 (1975), 427–433.

    MATH  Google Scholar 

  • 1976 Buxton, M. & Elmore, S. An extension of lower bounds for odd perfect numbers. Notices Amer. Math. Soc. 23 (1976), S. A55.

    Google Scholar 

  • 1976 Diffe, W. & Hellman, M.E. New directions in cryptography. IEEE Trans. on Inf. Th. IT-22 (1976), 644–654.

    Google Scholar 

  • 1976 Erdös, P. & Shorey, T.N. On the greatest prime factor of 2p − 1 for a prime p, and other expressions. Acta Arith. 30 (1976), 257–265.

    MATH  MathSciNet  Google Scholar 

  • 1976 Lehmer, D.H. Strong Carmichael numbers. J. Austral. Math. Soc. (A) 21 (1976), 508–510. Nachdruck in Selected Papers (Hrsg. D. McCarthy), Bd. I, 140–142. Ch. Babbage Res. Centre, St. Pierre, Manitoba 1981.

    Google Scholar 

  • 1976 Mendelsohn, N.S. The equation φ(x) = k. Math. Mag. 49 (1976), 37–39.

    MATH  MathSciNet  Google Scholar 

  • 1976 Miller, G.L. Riemann’s hypothesis and tests for primality. J. Comp. Syst. Sci. 13 (1976), 300–317.

    MATH  Google Scholar 

  • 1976 Rabin, M.O. Probabilistic algorithms. In Algorithms and Complexity (Hrsg. J.F. Traub), 21–39. Academic Press, New York 1976.

    Google Scholar 

  • 1976 Yorinaga, M. On a congruential property of Fibonacci numbers. Numerical experiments. Considerations and remarks. Math. J. Okayama Univ. 19 (1976), 5–10 und 11–17.

    MATH  MathSciNet  Google Scholar 

  • 1977 Kishore, M. Odd perfect numbers not divisible by 3 are divisible by at least ten distinct primes. Math. Comp. 31 (1977), 274–279.

    MATH  MathSciNet  Google Scholar 

  • 1977 Kishore, M. The number of distinct prime factors for which σ(N) = 2N, σ(N) = 2N ± 1 and φ(N) | N − 1. Dissertation, University of Toledo, Ohio 1977, 39 Seiten.

    Google Scholar 

  • 1977 Malm, D.E.G. On Monte-Carlo primality tests. Notices Amer. Math. Soc. 24 (1977), A-529, abstract 77T-A22.

    Google Scholar 

  • 1977 Pomerance, C. On composite n for which φ(n) | n − 1, II. Pacific J. Math. 69 (1977), 177–186.

    MathSciNet  Google Scholar 

  • 1977 Pomerance, C. Multiply perfect numbers, Mersenne primes and effective computability. Math. Ann. 226 (1977), 195–206.

    MATH  MathSciNet  Google Scholar 

  • 1977 Solovay, R. & Strassen, V. A fast Monte-Carlo test for primality. SIAM J. Comput. 6 (1977), 84–85.

    MATH  MathSciNet  Google Scholar 

  • 1977 Stewart, C.L. On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers. Proc. London Math. Soc. (3) 35 (1977), 425–447.

    Google Scholar 

  • 1977 Stewart, C.L. Primitive divisors of Lucas and Lehmer numbers. In Transcendence Theory: Advances and Applications (Hrsg. A. Baker und D.W. Masser), 79–92. Academic Press, London 1977.

    Google Scholar 

  • 1977 Williams, H.C. On numbers analogous to the Carmichael numbers. Can. Math. Bull. 20 (1977), 133–143.

    MATH  Google Scholar 

  • 1978 Cohen, G.L. On odd perfect numbers. Fibonacci Quart. 16 (1978), 523–527.

    MATH  MathSciNet  Google Scholar 

  • 1978 Kiss, P. & Phong, B.M. On a function concerning second order recurrences. Ann. Univ. Sci. Budapest 21 (1978), 119–122.

    MATH  Google Scholar 

  • 1978 Rivest, R.L. Remarks on a proposed cryptanalytic attack on the M.I.T. public-key cryptosystem. Cryptologia 2 (1978), 62–65.

    Google Scholar 

  • 1978 Rivest, R.L., Shamir, A. & Adleman, L.M. A method for obtaining digital signatures and public-key cryptosystems. Comm. ACM 21 (1978), 120–126.

    MATH  MathSciNet  Google Scholar 

  • 1978 Williams, H.C. Primality testing on a computer. Ars Comb. 5 (1978), 127–185.

    MATH  Google Scholar 

  • 1978 Yorinaga, M. Numerical computation of Carmichael numbers. Math. J. Okayama Univ. 20 (1978), 151–163.

    MATH  MathSciNet  Google Scholar 

  • 1979 Chein, E.Z. Non-existence of odd perfect numbers of the form \(q_1^{{a_1}}q_2^{{a_2}} \cdots q_6^{{a_6}}\,\,{\textrm{and}}\,\,{5^{{a_1}}}q_2^{{a_2}} \cdots q_9^{{a_9}}\). Dissertation, Pennsylvania State University, 1979.

    Google Scholar 

  • 1979 Lenstra Jr., H.W. Miller’s primality test. Inf. Process. Letters 8 (1979), 86–88.

    MATH  MathSciNet  Google Scholar 

  • 1980 Baillie, R. & Wagstaff Jr., S.S. Lucas pseudoprimes. Math. Comp. 35 (1980), 1391–1417.

    MATH  MathSciNet  Google Scholar 

  • 1980 Cohen, G.L. & Hagis Jr., P. On the number of prime factors of n if φ(n) | (n − 1). Nieuw Arch. Wisk. (3) 28 (1980), 177–185.

    Google Scholar 

  • 1980 Hagis Jr., P. Outline of a proof that every odd perfect number has at least eight prime factors. Math. Comp. 35 (1980), 1027–1032.

    MATH  MathSciNet  Google Scholar 

  • 1980 Monier, L. Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoret. Comput. Sci. 12 (1980), 97–108.

    MATH  MathSciNet  Google Scholar 

  • 1980 Pomerance, C., Selfridge, J.L. & Wagstaff Jr., S.S. The pseudoprimes to 25 · 109. Math. Comp. 35 (1980), 1003–1026.

    MATH  MathSciNet  Google Scholar 

  • 1980 Rabin, M.O. Probabilistic algorithm for testing primality. J. Number Theory 12 (1980), 128–138.

    MATH  MathSciNet  Google Scholar 

  • 1980 Wagstaff Jr., S.S. Large Carmichael numbers. Math. J. Okayama Univ. 22 (1980), 33–41.

    MATH  MathSciNet  Google Scholar 

  • 1980 Wall, D.W. Conditions for φ(N) to properly divide N−1. In A Collection of Manuscripts Related to the Fibonacci Sequence (Hrsg. V.E. Hoggatt und M. Bicknell-Johnson), 205–208. 18th Anniv. Vol., Fibonacci Assoc., San Jose 1980.

    Google Scholar 

  • 1980 Yorinaga, M. Carmichael numbers with many prime factors. Math. J. Okayama Univ. 22 (1980), 169–184.

    MATH  MathSciNet  Google Scholar 

  • 1981 Brent, R.P. & Pollard, J.M. Factorization of the eighth Fermat number. Math. Comp. 36 (1981), 627–630.

    MATH  MathSciNet  Google Scholar 

  • 1981 Grosswald, E. On Burgess’ bound for primitive roots modulo primes and an application to Γ(p). Amer. J. Math. 103 (1981), 1171–1183.

    MATH  MathSciNet  Google Scholar 

  • 1981 Hagis Jr., P. On the second largest prime divisor of an odd perfect number. In Analytic Number Theory (Hrsg. M.I. Knopp). Lecture Notes in Math. #899, 254–263. Springer-Verlag, New York 1981.

    Google Scholar 

  • 1981 Lenstra Jr., H.W. Primality testing algorithms (after Adleman, Rumely and Williams). In Séminaire Bourbaki, exposé No. 576. Lecture Notes in Math. #901, 243–257. Springer-Verlag, Berlin 1981.

    Google Scholar 

  • 1981 Lüneburg, H. Ein einfacher Beweis für den Satz von Zsigmondy über primitive Primteiler von A N − 1. In Geometries and Groups (Hrsg. M. Aigner und D. Jungnickel). Lecture Notes in Math. #893, 219–222. Springer-Verlag, New York 1981.

    Google Scholar 

  • 1982 Brent, R.P. Succinct proofs of primality for the factors of some Fermat numbers. Math. Comp. 38 (1982), 253–255.

    MATH  MathSciNet  Google Scholar 

  • 1982 Couvreur, C. & Quisquater, J.J. An introduction to fast generation of large prime numbers. Philips J. Res. 37 (1982), 231–264; Errata, 38 (1983), S. 77.

    Google Scholar 

  • 1982 Hoogendoorn, P.J. On a secure public-key cryptosystem. In Computational Methods in Number Theory (Hrsg. H.W. Lenstra, Jr. und R. Tijdeman), Teil I, 159–168. Math. Centre Tracts #154, Amsterdam 1982.

    Google Scholar 

  • 1982 Lenstra Jr., H.W. Primality testing. In Computational Methods in Number Theory (Hrsg. H.W. Lenstra, Jr. und R. Tijdeman), Teil I, 55–77. Math. Centre Tracts #154, Amsterdam 1982.

    Google Scholar 

  • 1982 Masai, P. & Valette, A. A lower bound for a counterexample in Carmichael’s conjecture. Boll. Un. Mat. Ital. (6) 1-A (1982), 313–316.

    Google Scholar 

  • 1982 Naur, T. Integer Factorization. DAIMI PB-144, Aarhus University, 1982, 129 Seiten.

    Google Scholar 

  • 1982 Woods, D. & Huenemann, J. Larger Carmichael numbers. Comput. Math. and Appl. 8 (1982), 215–216.

    MATH  MathSciNet  Google Scholar 

  • 1983 Adleman, L.M., Pomerance, C. & Rumely, R.S. On distinguishing prime numbers from composite numbers. Annals of Math. (2) 117 (1983), 173–206.

    Google Scholar 

  • 1983 Brillhart, J., Lehmer, D.H., Selfridge, J.L., Tuckerman, B. & Wagstaff Jr., S.S. Factorizations of b n ±1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers. Contemporary Math., Bd. 22, Amer. Math. Soc., Providence, RI 1983 (2. Auflage 1988, 3. Auflage 2002 nur elektronisch).

    Google Scholar 

  • 1983 Hagis Jr., P. Sketch of a proof that an odd perfect number relatively prime to 3 has at least eleven prime factors. Math. Comp. 40 (1983), 399–404.

    MATH  MathSciNet  Google Scholar 

  • 1983 Kishore, M. Odd perfect numbers not divisible by 3, II. Math. Comp. 40 (1983), 405–411.

    MATH  MathSciNet  Google Scholar 

  • 1983 Naur, T. New integer factorizations. Math. Comp. 41 (1983), 687–695.

    MATH  MathSciNet  Google Scholar 

  • 1983 Pomerance, C. & Wagstaff Jr., S.S. Implementation of the continued fraction integer factoring algorithm. Congressus Numerantium 37 (1983), 99–118.

    MathSciNet  Google Scholar 

  • 1983 Powell, B. Problem 6420 (On primitive roots). Amer. Math. Monthly 90 (1983), S. 60.

    MathSciNet  Google Scholar 

  • 1983 Singmaster, D. Some Lucas pseudoprimes. Abstracts Amer. Math. Soc. 4 (1983), abstract 83T-10-14, S. 197.

    Google Scholar 

  • 1983 Yates, S. Titantic primes. J. Recr. Math. 16 (1983/84), 250–260.

    MathSciNet  Google Scholar 

  • 1984 Cohen, H. & Lenstra Jr., H.W. Primality testing and Jacobi sums. Math. Comp. 42 (1984), 297–330.

    MATH  MathSciNet  Google Scholar 

  • 1984 Dixon, J.D. Factorization and primality tests. Amer. Math. Monthly 91 (1984), 333–352.

    MATH  Google Scholar 

  • 1984 Kearnes, K. Solution of problem 6420. Amer. Math. Monthly 91 (1984), S. 521.

    MathSciNet  Google Scholar 

  • 1984 Nicolas, J.L. Tests de primalité. Expo. Math. 2 (1984), 223–234.

    MATH  MathSciNet  Google Scholar 

  • 1984 Pomerance, C. Lecture Notes on Primality Testing and Factoring (Mitschrift von G.M. Gagola, Jr.). Math. Assoc. America, Notes No. 4, 1984, 34 Seiten.

    Google Scholar 

  • 1984 Williams, H.C. An overview of factoring. In Advances in Cryptology (Hrsg. D. Chaum), 71–80. Plenum, New York 1984.

    Google Scholar 

  • 1984 Yates, S. Sinkers of the Titanics. J. Recr. Math. 17 (1984/85), 268–274.

    MathSciNet  Google Scholar 

  • 1985 Bedocchi, E. Note on a conjecture on prime numbers. Rev. Mat. Univ. Parma (4) 11 (1985), 229–236.

    Google Scholar 

  • 1985 Fouvry, E. Théorème de Brun-Titchmarsh; application au théor ème de Fermat. Invent. Math. 79 (1985), 383–407.

    MATH  MathSciNet  Google Scholar 

  • 1985 Riesel, H. Prime Numbers and Computer Methods for Factorization. Birkhäuser, Boston 1985 (2. Auflage 1994).

    MATH  Google Scholar 

  • 1985 Shan, Z. On composite n for which φ(n) | n−1. J. China Univ. Sci. Technol. 15 (1985), 109–112.

    MATH  MathSciNet  Google Scholar 

  • 1985 Subbarao, M.V. & Siva Rama Prasad, V. Some analogues of a Lehmer problem on the totient function. Rocky Mountain J. Math. 15 (1985), 609–620.

    MATH  MathSciNet  Google Scholar 

  • 1985 Wagon, S. Perfect numbers. Math. Intelligencer 7, No. 2 (1985), 66–68.

    MATH  MathSciNet  Google Scholar 

  • 1986 Kiss, P., Phong, B.M. & Lieuwens, E. On Lucas pseudoprimes which are products of s primes. In Fibonacci Numbers and their Applications (Hrsg. A.N. Philippou, G.E. Bergum und A.F. Horadam), 131–139. Reidel, Dordrecht 1986.

    Google Scholar 

  • 1986 Wagon, S. Carmichael’s “Empirical Theorem”. Math. Intelligencer 8, No. 2 (1986), 61–62.

    MATH  MathSciNet  Google Scholar 

  • 1986 Wagon, S. Primality testing. Math. Intelligencer 8, No. 3 (1986), 58–61.

    MATH  MathSciNet  Google Scholar 

  • 1987 Cohen, G.L. On the largest component of an odd perfect number. J. Austral. Math. Soc. (A) 42 (1987), 280–286.

    Google Scholar 

  • 1987 Cohen, H. & Lenstra, A.K. Implementation of a new primality test. Math. Comp. 48 (1987), 103–121 und S1–S4.

    MathSciNet  Google Scholar 

  • 1987 Koblitz, N. A Course in Number Theory and Cryptography. Springer-Verlag, New York 1987 (2. Auflage 1994).

    MATH  Google Scholar 

  • 1987 Lenstra Jr., H.W. Factoring integers with elliptic curves. Annals of Math. 126 (1987), 649–673.

    MathSciNet  Google Scholar 

  • 1987 Li Yan & Du Shiran Chinese Mathematics. A Concise History (Englische übersetzung von J.N. Crossley und A.W.C. Lun). Clarendon Press, Oxford 1987.

    Google Scholar 

  • 1987 Pomerance, C. Very short primality proofs. Math. Comp. 48 (1987), 315–322.

    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.

    MathSciNet  Google Scholar 

  • 1988 Hagis Jr., P. On the equation Mφ(n) = n − 1. Nieuw Arch. Wisk. (4) 6 (1988), 237–243.

    Google Scholar 

  • 1988 Young, J. & Buell, D.A. The twentieth Fermat number is composite. Math. Comp. 50 (1988), 261–263.

    MATH  MathSciNet  Google Scholar 

  • 1989 Bateman, P.T., Selfridge, J.L. & Wagstaff Jr., S.S. The new Mersenne conjecture. Amer. Math. Monthly 96 (1989), 125–128.

    MATH  MathSciNet  Google Scholar 

  • 1989 Brent, R.P. & Cohen, G.L. A new lower bound for odd perfect numbers. Math. Comp. 53 (1989), 431–437 und S7–S24.

    MathSciNet  Google Scholar 

  • 1989 Bressoud, D.M. Factorization and Primality Testing. Springer-Verlag, New York 1989.

    MATH  Google Scholar 

  • 1989 Dubner, H. A new method for producing large Carmichael numbers. Math. Comp. 53 (1989), 411–414.

    MATH  MathSciNet  Google Scholar 

  • 1989 Lemos, M. Criptografia, Núumeros Primos e Algoritmos. 17° Colóquio Brasileiro de Matemática, Inst. Mat. Pura e Aplic., Rio de Janeiro, 1989, 72 Seiten.

    Google Scholar 

  • 1990 Lenstra, A.K. & Lenstra Jr., H.W. Algorithms in number theory. In Handbook of Theoretical Computer Science (Hrsg. J. van Leeuwen, A. Meyer, M. Nivat, M. Paterson und D. Perrin). North-Holland, Amsterdam 1990.

    Google Scholar 

  • 1991 Brent, R.P., Cohen, G.L. & te Riele, H.J.J. Improved techniques for lower bounds for odd perfect numbers. Math. Comp. 57 (1991), 857–868.

    MATH  MathSciNet  Google Scholar 

  • 1991 Frasnay, C. Extension à l’anneau ℤ p du théorème de Lucas, sur les coefficients binomiaux. Singularité 2 (1991), 13–15.

    MATH  Google Scholar 

  • 1992 Yates, S. Collecting gigantic and titanic primes. J. Recr. Math. 24 (1992), 187–195.

    Google Scholar 

  • 1993 Atkin, A.O.L. & Morain, F. Elliptic curves and primality proving. Math. Comp. 61 (1993), 29–68.

    MATH  MathSciNet  Google Scholar 

  • 1993 Cohen, H. A Course in Computational Algebraic Number Theory. Springer-Verlag, New York 1993.

    MATH  Google Scholar 

  • 1993 Jaeschke, G. On strong pseudoprimes to several bases. Math. Comp. 61 (1993), 915–926.

    MATH  MathSciNet  Google Scholar 

  • 1993 Lenstra, A.K., Lenstra Jr., H.W., Manasse, M.S. & Pollard, J.M. The number field sieve. In The Development of the Number Field Sieve (Hrsg. A.K. Lenstra und H.W. Lenstra, Jr.). Lecture Notes in Math. #1554, 11–42. Springer-Verlag, New York 1993.

    Google Scholar 

  • 1993 Pomerance, C. Carmichael numbers. Nieuw Arch. Wisk. (4) 11 (1993), 199–209.

    Google Scholar 

  • 1993 Williams, H.C. How was F 6 factored? Math. Comp. 61 (1993), 463–474.

    MATH  MathSciNet  Google Scholar 

  • 1994 Alford, W.R., Granville, A. & Pomerance, C. There are infinitely many Carmichael numbers. Annals of Math. (2) 140 (1994), 703–722.

    Google Scholar 

  • 1994 Dubner, H. Palindromic primes with a palindromic prime number of digits. J. Recr. Math. 26 (1994), 241–243.

    Google Scholar 

  • 1994 Heath-Brown, D.R. Odd perfect numbers. Math. Proc. Cambridge Phil. Soc. 115 (1994), 191–196.

    MATH  MathSciNet  Google Scholar 

  • 1994 Schlay, A. & Wagon, S. Carmichael’s conjecture on the Euler function is valid below 1010000000. Math. Comp. 63 (1994), 415–419.

    MathSciNet  Google Scholar 

  • 1994 Williams, H.C. & Shallit, J.O. Factoring integers before computers. In Mathematics of Computation, 1943-1993: A Half- Century of Computational Mathematics (Hrsg. W. Gautschi). Proc. Symp. Appl. Math., Bd. 48, 481–531. Amer. Math. Soc., Providence, RI 1994.

    Google Scholar 

  • 1995 Crandall, R.E., Doenias, J., Norrie, C. & Young, J. The twenty-second Fermat number is composite. Math. Comp. 64 (1995), 863–868.

    MATH  MathSciNet  Google Scholar 

  • 1995 Gostin, G.B. New factors of Fermat numbers. Math. Comp. 64 (1995), 393–395.

    MATH  MathSciNet  Google Scholar 

  • 1995 McIntosh, R.J. On the converse of Wolstenholme’s theorem. Acta Arith. 71 (1995), 381–389.

    MATH  MathSciNet  Google Scholar 

  • 1995 Trevisan, V. & Carvalho, J.B. The composite character of the twenty-second Fermat number. J. Supercomp. 9 (1995), 179–182.

    MATH  Google Scholar 

  • 1996 Borwein, D., Borwein, J.M., Borwein, P.B. & Girgensohn, R. Giuga’s conjecture on primality. Amer. Math. Monthly 103 (1996), 40–50.

    MATH  MathSciNet  Google Scholar 

  • 1996 Löh, G. & Niebuhr W. A new algorithm for constructing large Carmichael numbers. Math. Comp. 65 (1996), 823–836.

    MATH  MathSciNet  Google Scholar 

  • 1996 Pomerance, C. A tale of two sieves. Notices Amer. Math. Soc. 43 (1996), 1473–1485.

    MATH  MathSciNet  Google Scholar 

  • 1997 Cohen, G.L. Numbers whose positive divisors have small integral harmonic mean. Math. Comp. 66 (1997), 883–891.

    MATH  MathSciNet  Google Scholar 

  • 1998 Caldwell, C. & Dubner, H. Primes in π. J. Recr. Math. 29 (1998), 282–289.

    Google Scholar 

  • 1998 Ford, K. The distribution of totients. Hardy-Ramanujan J. 2 (1998), 67–151.

    MATH  Google Scholar 

  • 1998 Hagis Jr., P. & Cohen, G.L. Every odd perfect number has a prime factor which exceeds 106. Math. Comp. 67 (1998), 1323–1330.

    MATH  MathSciNet  Google Scholar 

  • 1998 Williams, H.C. Édouard Lucas and Primality Testing. J. Wiley and Sons, New York 1998.

    MATH  Google Scholar 

  • 1998 Young, J. Large primes and Fermat factors. Math. Comp. 67 (1998), 1735–1738.

    MATH  MathSciNet  Google Scholar 

  • 1999 Brent, R.P. Factorization of the tenth Fermat number. Math. Comp. 68 (1999), 429–451.

    MATH  MathSciNet  Google Scholar 

  • 1999 Cook, R.J. Bounds for odd perfect numbers. In Number Theory (Hrsg. R. Gupta). CRM Proc. Lecture Notes #19, 67–71. Amer. Math. Soc., Providence, RI 1999.

    Google Scholar 

  • 1999 Coutinho, S.C. The Mathematics of Ciphers: Number Theory and RSA Cryptography. A.K. Peters, Natick, MA 1999.

    MATH  Google Scholar 

  • 1999 Ford, K. The number of solutions of φ(x) = m. Annals of Math. (2) 150 (1999), 283–311.

    Google Scholar 

  • 1999 Grytczuk, A. & Wojtowicz, M. There are no small odd perfect numbers, and Erratum. C.R. Acad. Sci. Paris 328 (1999), 1101–1105, und 330 (2000), S. 533.

    MATH  MathSciNet  Google Scholar 

  • 1999 Iannucci, D.E. The second largest prime divisor of an odd perfect number exceeds ten thousand. Math. Comp. 68 (1999), 1749–1760.

    MATH  MathSciNet  Google Scholar 

  • 1999 Woltman, G.F. On the discovery of the 38th known Mersenne prime. Fibonacci Quart. 37 (1999), 367–370.

    MATH  MathSciNet  Google Scholar 

  • 2000 Brent, R.P., Crandall, R.E., Dilcher, K. & van Halewyn, C. Three new factors of Fermat numbers. Math. Comp. 69 (2000), 1297–1304.

    MATH  MathSciNet  Google Scholar 

  • 2000 Iannucci, D.E. The third largest prime divisor of an odd perfect number exceeds one hundred. Math. Comp. 69 (2000), 867–879.

    MATH  MathSciNet  Google Scholar 

  • 2001 Crandall, R. & Pomerance, C. Prime Numbers. A Computational Perspective. Springer-Verlag, New York 2001 (2. Auflage 2005).

    Google Scholar 

  • 2001 Křížek, M., Luca, F. & Somer, L. 17 Lectures on Fermat Numbers. From Number Theory to Geometry. Springer-Verlag, New York 2001.

    MATH  Google Scholar 

  • 2001 Ribenboim, P. Classical Theory of Algebraic Numbers. Springer- Verlag, New York 2001.

    MATH  Google Scholar 

  • 2001 Zhang, Z. Finding strong pseudoprimes to several bases. Math. Comp. 70 (2001), 863–872.

    MATH  MathSciNet  Google Scholar 

  • 2002 Agrawal, M., Kayal, N. & Saxena, N. PRIMES is in P. Unveröffentlichtes Manuskript.

    Google Scholar 

  • 2002 Morain, F. Primalité théorique et primalité pratique ou AKS vs. ECCP. Unveröffentlichtes Manuskript.

    Google Scholar 

  • 2003 Bailey, D.H. Some background on Kanada’s recent pi calculation. Unveröffentlichtes Manuskript.

    Google Scholar 

  • 2003 Crandall, R.E., Mayer, E.W. & Papadopoulos, J.S. The twenty-fourth Fermat number is composite. Math. Comp. 72 (2003), 1555–1572.

    MATH  MathSciNet  Google Scholar 

  • 2003 Jenkins, P.M. Odd perfect numbers have a prime factor exceeding 107. Math. Comp. 72 (2003), 1549–1554.

    MATH  MathSciNet  Google Scholar 

  • 2003 Nielsen, P.P. An upper bound for odd perfect numbers. Integers 3 (2003), #A14, 1–9 (elektronisch).

    Google Scholar 

  • 2003 Sorli, R.M. Algorithms in the study of multiperfect and odd perfect numbers. Dissertation, University of Technology, Sydney 2003.

    Google Scholar 

  • 2003 Wagstaff Jr., S.S. Cryptanalysis of Number Theoretic Ciphers. Chapman & Hall /CRC, Boca Raton, FL 2003.

    MATH  Google Scholar 

  • 2003 Zhang, Z. & Tang, M. Finding strong pseudoprimes to several bases. II. Math. Comp. 72 (2003), 2085–2097.

    MATH  MathSciNet  Google Scholar 

  • 2004 Agrawal, M., Kayal, N. & Saxena, N. PRIMES is in P. Annals of Math. 160 (2004), 781–793.

    MATH  MathSciNet  Google Scholar 

  • 2005 Lenstra Jr., H.W. & Pomerance, C. Primality testing with Gaussian periods. Unveröffentlichtes Manuskript.

    Google Scholar 

  • 2005 Morain, F. Implementing the asymptotically fast version of the Elliptic Curve Primality Proving algorithm. Preprint arXiv:math. NT/0502097, 2005.

    Google Scholar 

  • 2007 Goto, T. & Okeya, K. All harmonic numbers less than 1014. Japan J. Indust. Appl. Math. 24 (2007), 275–288.

    MATH  MathSciNet  Google Scholar 

  • 2007 McIntosh, R.J. & Roettger, E.L. A search for Fibonacci- Wieferich and Wolstenholme primes. Math. Comp. 76 (2007), 2087–2094.

    MATH  MathSciNet  Google Scholar 

  • 2007 Nielsen, P.P. Odd perfect numbers have at least nine distinct prime factors. Math. Comp. 76 (2007), 2109–2126.

    MATH  MathSciNet  Google Scholar 

  • 2007 Pinch, R.G.E. The Carmichael Numbers up to 1021. In Proc. Conf. on Algorithmic Number Th. 2007 (Hrsg. A.-M. Ernvall-Hytönen, M. Jutila, J. Karhumäki und A. Lepistö), 129–131. Turku Centre for Computer Science, Turku 2007.

    Google Scholar 

  • 2008 Goto, T. & Ohno, Y. Odd perfect numbers have a prime factor exceeding 108. Math. Comp. 77 (2008), 1859–1868.

    MATH  MathSciNet  Google Scholar 

  • 2010 Cohen, G.L. & Sorli, R.M. Odd harmonic numbers exceed 1024. Math. Comp. 79 (2010), 2451–2460.

    MATH  MathSciNet  Google Scholar 

  • Keller, W. Primfaktoren k · 2n + 1 von Fermat-Zahlen F m und vollständiger Faktorisierungsstatus. http://www.prothsearch.net/fermat.html

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

  • Woltman, G. Status der Great Internet Mersenne Prime Search. http://www.mersenne.org/report_milestones/

  • Caldwell, C. Status der Mersenne-Vermutung von Bateman, Selfridge und Wagstaff. http://primes.utm.edu/mersenne/NewMersenneConjecture.html

  • Wagstaff Jr., S.S. Das Cunningham-Projekt. http://homes.cerias.purdue.edu/~ssw/cun/

  • Martin, M. Größte mit dem ECPP-Verfahren verifizierte Primzahlen. http://www.ellipsa.net/primo/top20.html

  • Kelly, B. Faktorisierung von Fibonacci- und Lucas-Zahlen. http://mersennus.net/fibonacci/index.html

  • Contini, S. Rekord-Faktorisierungen. http://www.crypto-world.com/FactorAnnouncements.html

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    Prof. Dr. Paulo Ribenboim

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Ribenboim, P. (2011). Wie kann man Primzahlen erkennen?. In: Die Welt der Primzahlen. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18079-8_2

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