Advertisement

How Are the Prime Numbers Distributed?

  • Paulo Ribenboim

Abstract

As I have already stressed, the various proofs of existence of infinitely many primes are not constructive and do not give an indication of how to determine the nth prime number The proofs also do not indicate how many primes are less than any given number N. By the same token, there is no reasonable formula or function representing primes.

Keywords

Prime Number Zeta Function Arithmetic Progression Critical Line Riemann Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1885.
    MEISSEL, E.D.F. Berechnung der Menge von Primzahlen, welche innerhalb der ersten Milliarde natürlicher Zahlen vorkommen. Math. Ann., 25, 1885, 251–257.MathSciNetGoogle Scholar
  2. 1892.
    SYLVESTER, J.J. On arithmetical series. Messenger of Math., 21, 1892, 1–19 and 87–120. Reprinted inGesammelte Abhandlungen, Vol. III, 573–587. Springer-Verlag, New York, 1968.Google Scholar
  3. 1901.
    VON KOCH, H. Sur la distribution des nombres premiers. Acta Math., 24, 1901, 159–182.MathSciNetGoogle Scholar
  4. 1901.
    WOLFSKEHL, P. Ueber eine Aufgabe der elementaren Arithmetik. Math. Ann., 54, 1901, 503–504.MathSciNetzbMATHGoogle Scholar
  5. 1909.
    LANDAU, E. Handbuch der Lehre von der Verteilung der Primzahlen. Teubner, Leipzig, 1909. Reprinted by Chelsea, Bronx, N.Y., 1974.Google Scholar
  6. 1914.
    LITTLEWOOD, J.E. Sur la distribution des nombres premiers. C.R. Acad. Sci. Paris, 158, 1914, 869–872.Google Scholar
  7. 1919.
    BRUN, V. Le crible d’Eratosthène et le théorème de Goldbach. C.R. Acad. Sci. Paris, 168, 1919, 544–546.zbMATHGoogle Scholar
  8. 1919.
    BRUN, V. La série 5+7+11+13+17+19+29+31+ 41 + 43 + 59 + 61 +... où les dénominateurs sont “nombres premiers jumeaux” est convergente ou finie. Bull. Sci. Math., (2), 43, 1919, 100–104 and 124–128.Google Scholar
  9. 1920.
    BRUN, V. Le crible d’Erathostène et la théorème de Goldbach. Videnskapsselskapets Skrifter Kristiania, Mat.-nat. Kl. 1920, No. 3, 36 pages.Google Scholar
  10. 1923.
    HARDY, G. H. & LITTLEWOOD, J.E. Some problems of “Partitio Numerorum”, III: On the expression of a number as a sum of primes. Acta Math., 44, 1923, 1–70. Reprinted inCollected Papers of G.H. Hardy, Vol. I, 561–630. Clarendon Press, Oxford, 1966.Google Scholar
  11. 1930.
    ERDOS, P. Beweis eines Satzes von Tschebycheff. Acta Sci. Math. Szeged, 5, 1930, 194–198.Google Scholar
  12. 1930.
    HOHEISEL, G. Primzahlprobleme in der Analysis. Sitzungsberichte Berliner Akad. d. Wiss., 1930, 580–588.Google Scholar
  13. 1930.
    SCHNIRELMANN, L. Uber additive Eigenschaften von Zahlen. Ann. Inst. Polytechn. Novocerkask, 14, 1930, 3–28 and Math. Ann., 107, 1933, 649–690.Google Scholar
  14. 1933.
    SKEWES, S. On the differencer(x) — li(x). J. London Math. Soc., 8, 1933, 277–283.MathSciNetGoogle Scholar
  15. 1934.
    ISHIKAWA, H. Uber die Verteilung der Primzahlen. Sci. Rep. Tokyo Bunrika Daigaku, A, 2, 1934, 27–40.Google Scholar
  16. 1937.
    CRAMÉR, H. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithm., 2, 1937, 2346.Google Scholar
  17. 1937.
    INGHAM, primes. Quart. 255–266.Google Scholar
  18. 1937.
    LANDAU, A.E. On the difference between consecutive J. Pure Si Appl. Math., Oxford, Ser. 2, 8, 1937, E. Über einige neuere Fortschritte der additiven Zahlentheorie. Cambridge Univ. Press, Cambridge, 1937. Reprinted by Stechert-Hafner, New York, 1964.Google Scholar
  19. 1937.
    VAN DER CORPUT, J.G. Sur l’hypothèse de Goldbach pour presque tous les nombres pairs. Acta Arithm., 2, 1937, 266–290.Google Scholar
  20. 1937.
    VINOGRADOV, I.M. Representation of an odd number as the sum of three primes (in Russian). Dokl. Akad. Nauk SSSR, 15, 1937, 169–172.zbMATHGoogle Scholar
  21. 1938.
    ESTERMANN, T. Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc., 44, 1938, 307–314.Google Scholar
  22. 1938.
    POULET, P. Table des nombres composés vérifiant le théorème de Fermat pour le module 2, jusqu’ à 100.000.000. Sphinx, 8, 1938, 52–52 Corrections: Math. Comp., 25, 1971, 944–945 and 26, 1972, p. 814.Google Scholar
  23. 1938.
    ROSSER, J.B. The nth prime is greater than n log n. Proc. London Math. Soc. 45, 1938, 21–44.Google Scholar
  24. 1938.
    TSCHUDAKOFF, N.G. On the density of the set of even integers which are not representable as a sum of two odd primes (in Russian). Izv. Akad. Nauk SSSR, Ser. Mat., 1, 1938, 25–40.Google Scholar
  25. 1939.
    VAN DER CORPUT, J.G. Über Summen von Primzahlen und Prim-zahlquadraten. Math. Ann., 116, 1939, 1–50.MathSciNetGoogle Scholar
  26. 1944.
    LINNIK, Yu.V. On the least prime in an arithmetic progression I. The basic theorem (in Russian). Mat. Sbornik, 15 (57), 1944, 139–178.MathSciNetGoogle Scholar
  27. 1946.
    BRAUER, A. On the exact number of primes below a given limit. Amer. Math. Monthly, 9, 1946, 521–523.MathSciNetGoogle Scholar
  28. 1947.
    KHINCHIN, A.Ya. Three Pearls of Number Theory. Original Russian edition in OGIZ, Moscow, 1947. Translation into English published by Graylock Press, Baltimore, 1952.Google Scholar
  29. 1947.
    RÉNYI, A. On the representation of even numbers as the sum of a prime and an almost prime. Dokl. Akad. Nauk SSSR, 56, 1947, 455–458.zbMATHGoogle Scholar
  30. 1949.
    CLEMENT, P.A. Congruences for sets of primes. Amer. Math. Monthly, 56, 1949, 23–25.MathSciNetzbMATHGoogle Scholar
  31. 1949.
    ERDÖS, P. On a new method in elementary number theory which leads to an elementary proof of the prime number theorem. Proc. Nat. Acad. Sci. U.S.A., 35, 1949, 374–384.MathSciNetzbMATHGoogle Scholar
  32. 1949.
    MOSER, L. A theorem on the distribution of primes. Amer. Math. Monthly, 56, 1949, 624–625.zbMATHGoogle Scholar
  33. 1949.
    RICHERT, H.E. Über Zerfällungen in ungleiche Primzahlen. Math. Zeits., 52, 1949, 342–343.MathSciNetzbMATHGoogle Scholar
  34. 1949.
    SELBERG, A. An elementary proof of the prime number theorem. Annals of Math., 50, 1949, 305–313.MathSciNetzbMATHGoogle Scholar
  35. 1949.
    SELBERG, A. An elementary proof of Dirichlet’s theorem about primes in an arithmetic progression. Annals of Math., 50, 1949, 297–304.MathSciNetzbMATHGoogle Scholar
  36. 1949.
    SELBERG, A. An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math., 2, 1950, 66–78.MathSciNetzbMATHGoogle Scholar
  37. 1950.
    ERDÖS, P. On almost primes. Amer. Math. Monthly, 57, 1950, 404–407.MathSciNetzbMATHGoogle Scholar
  38. 1950.
    HASSE, H. Vorlesungen über Zahlentheorie. Springer-Verlag, Berlin, 1950.zbMATHGoogle Scholar
  39. 1950.
    SELBERG, A. The general sieve method and its place in prime number theory. Proc. Int. Congr. Math., Cambridge, 1950.Google Scholar
  40. 1951.
    TITCHMARSH, E.C. The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951.zbMATHGoogle Scholar
  41. 1956.
    ERDOS, P. On pseudo-primes and Carmichael numbers. Publ. Math. Debrecen, 4, 1956, 201–206.MathSciNetGoogle Scholar
  42. 1957.
    LEECH, J. Note on the distribution of prime numbers. J. London Math. Soc., 32, 1957, 56–58.MathSciNetzbMATHGoogle Scholar
  43. 1958.
    SCHINZEL, A. & SIERPINSKI, W. Sur certaines hypothèses concernant les nombres premiers. Acta Arithm., 4, 1958, 185–208; Erratum, 5, 1959, p. 259.MathSciNetGoogle Scholar
  44. 1959.
    SCHINZEL, A. Démonstration d’une conséquence de l’hypothèse de Goldbach. Compositio Math., 14, 1959, 74–76.MathSciNetGoogle Scholar
  45. 1961.
    WRENCH, J.W. Evaluation of Artin’s constant and the twin-prime constant. Math. Comp., 15, 1961, 396–398.MathSciNetzbMATHGoogle Scholar
  46. 1962.
    ROSSER, J.B. & SCHOENFELD, L. Approximate formulas for some functions of prime numbers. Illinois J. Math., 6, 1962, 64–94.MathSciNetzbMATHGoogle Scholar
  47. 1963.
    AYOUB, R.G. AnIntroduction to the Theory of Numbers. Amer. Math. Soc., Providence, R.I., 1963.Google Scholar
  48. 1963.
    KANOLD, H.J. Elementare Betrachtungen zur Primzahltheorie. Arch. Math., 14, 1963, 147–151.MathSciNetzbMATHGoogle Scholar
  49. 1963.
    ROTKIEWICZ, A. Sur les nombres pseudo-premiers de la formeax + b. C.R. Acad. Sci. Paris, 257, 1963, 2601–2604.MathSciNetzbMATHGoogle Scholar
  50. 1963.
    WALFISZ, A.Z. Weylsche Exponentialsummen in der neueren Zahfentheorie. VEB Deutscher Verlag d. Wiss., Berlin, 1963.Google Scholar
  51. 1965.
    GELFOND, A.O. & LINNIK, Yu.V. Elementary Methods in Analytic Number Theory. Translated by A. Feinstein, revised and edited by L.J. Mordell. Rand McNally, Chicago, 1965.Google Scholar
  52. 1965.
    PAN, C.D. On the least prime in an arithmetic progression. Sci. Record (N.S.), 1, 1957, 311–313.MathSciNetzbMATHGoogle Scholar
  53. 1965.
    ROTKIEWICZ, A. Les intervalles contenant les nombres pseudo premiers. Rend. Circ. Mat. Palermo (2), 14, 1965, 278–280.MathSciNetzbMATHGoogle Scholar
  54. 1965.
    STEIN, M.L.& STEIN, P.R. New experimental results on the Goldbach conjecture. M.th. Mag., 38, 1965, 72–80.Google Scholar
  55. 1965.
    STEIN, M.L. & STEIN, P.R. Experimental results on additive 2-bases. Math. Comp., 19, 1965, 427–434.zbMATHGoogle Scholar
  56. 1966.
    BOMBIERI, E. & DAVENPORT, H. Small differences between prime numbers. Proc. Roy. Soc., A, 293, 1966, 1–18.MathSciNetzbMATHGoogle Scholar
  57. 1967.
    LANDER, L.J. & PARKIN, T.R. Consecutive primes in arithmetic progression. Math. Comp., 21, 1967, p. 489.zbMATHGoogle Scholar
  58. 1967.
    ROTKIEWICZ, A. On the pseudo-primes of the formax + b. Proc. Cambridge Phil. Soc., 63, 1967, 389–392.MathSciNetzbMATHGoogle Scholar
  59. 1967.
    SZYMICZEK, K. On pseudo-primes which are products of distinct primes. Amer. Math. Monthly, 74, 1967, 35–37.MathSciNetzbMATHGoogle Scholar
  60. 1969.
    MONTGOMERY, H.L. Zeros of L-functions. Invent. Math., 8, 1969, 346–354.zbMATHGoogle Scholar
  61. 1969.
    ROSSER, J.B., YOHE, J.M. & SCHOENFELD, L. Rigorous computation of the zeros of the Riemann zeta-function (with discussion). Inform. Processing 68 (Proc. IFIP Congress, Edinburgh, 1968 ), Vol. I, 70–76. North-Holland, Amsterdam, 1969.Google Scholar
  62. 1971.
    TITCHMARSH, E.C. The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951.zbMATHGoogle Scholar
  63. 1972.
    HUXLEY, M.N. On the difference between consecutive primes. Invent. Math., 15, 1972, 164–170.MathSciNetzbMATHGoogle Scholar
  64. 1972.
    HUXLEY, M.N. The Distribution of Prime Numbers. Oxford Univ. Press, Oxford, 1972.Google Scholar
  65. 1972.
    ROTKIEWICZ, A. On a problem of W. Sierpinski. Elem. d. Math., 27, 1972, 83–85.MathSciNetzbMATHGoogle Scholar
  66. 1973/1978.
    CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II. Sci. Sinica, 16, 1973, 157–176;Google Scholar
  67. 1973/1978.
    CHEN, J.R. On the representation of a large even integer as the sum of a prime and the product of at most two primes, I and II. Sci. Sinica, 21, 1978, 421–430.Google Scholar
  68. 1973.
    MONTGOMERY, H.L. The pair correlation of zeros of the zeta function. Analytic Number Theory (Proc. Symp. Pure Math., Vol. XXIV, St. Louis, 1972 ), 181–193. Amer. Math. Soc., Providence, R.I., 1973.Google Scholar
  69. 1974.
    AYPUB, R.B. Euler and the zeta-function. Amer. Math. Monthly, 81, 1974, 1067–1086.Google Scholar
  70. 1974.
    EDWARDS, H.M. Riemann’s Zeta Function. Academic Press, New York, 1974.zbMATHGoogle Scholar
  71. 1974.
    HALBERSTAM, H. & RICHERT, H.E. Sieve Methods. Academic Press, New York, 1974.zbMATHGoogle Scholar
  72. 1974.
    LEVINSON, N. More than one third of zeros of Riemann’s zeta function are ona = 1/2. Adv. in Math., 13, 1984, 383–436.MathSciNetGoogle Scholar
  73. 1974.
    MAKOWSKI, A. On a problem of Rotkiewicz on pseudoprimes. Elem. d. Math., 29, 1974, p. 13.Google Scholar
  74. 1975.
    MONTGOMERY, H.L. & VAUGHAN, R.C. The exceptional set in Goldbach’s problem. Acta Arithm., 27, 1975, 353–370.MathSciNetzbMATHGoogle Scholar
  75. 1975.
    ROSS, P.M. On Chen’s theorem that each large even number has the formp i + p2 or pi + p2p3. J. London Math. Soc., (2), 10, 1975, 500–506.MathSciNetzbMATHGoogle Scholar
  76. 1975.
    ROSSER, J.B. & SCHOENFELD, L. Sharper bounds for Chebyshev functions0(x) and1(x). Math. Comp., 29, 1975, 243–269.MathSciNetzbMATHGoogle Scholar
  77. 1976.
    APOSTOL, T.M. Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976.zbMATHGoogle Scholar
  78. 1976.
    BRENT, R.P. Tables concerning irregularities in the distribution of primes and twin primes to 1011. Math. Comp., 30, 1976, p. 379.MathSciNetGoogle Scholar
  79. 1977.
    HUDSON, R.H. A formula for the exact number of primes below a given bound in any arithmetic progression. Bull. Austral. Math. Soc., 16, 1977, 67–73.MathSciNetzbMATHGoogle Scholar
  80. 1977.
    HUDSON, R.H. & BRAUER, A. On the exact number of primes in the arithmetic progressions 4n + 1 and 6n + 1. Journal f. d. reine u. angew. Math., 291, 1977, 23–29.MathSciNetzbMATHGoogle Scholar
  81. 1977.
    LANGEVIN, M. Méthodes élémentaires en vue du théorème de Sylvester. Sém. Delange-Pisot-Poitou, 17e année, 1975/76, fasc. 1, exp. No. G12, 9 pages, Paris, 1977.Google Scholar
  82. 1977.
    WEINTRAUB, S. Seventeen primes in arithmetic progression. Math. Comp., 31, 1977, p. 1030.MathSciNetzbMATHGoogle Scholar
  83. 1978.
    BAYS, C.& HUDSON, R.H. On the fluctuations of Littlewood for primes of the form 4n + 1. Math. Comp., 32, 141, 281–286.Google Scholar
  84. 1978.
    HEATH-BROWN, D.R. Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Phil. Soc., 83, 1978, 357–375.MathSciNetzbMATHGoogle Scholar
  85. 1979.
    HEATH-BROWN, D.R. & IWANIEC, H. On the difference between consecutive powers. Bull. Amer. Math. Soc., N.S., 1, 1979, 758–760.MathSciNetzbMATHGoogle Scholar
  86. 1979.
    IWANIEC, H. & JUTILA, M. Primes in short intervals. Arkiv f. Mat., 17, 1979, 167–176.MathSciNetzbMATHGoogle Scholar
  87. 1979.
    POMERANCE, C. The prime number graph. Math. Comp., 33, 1979, 399–408.MathSciNetzbMATHGoogle Scholar
  88. 1979.
    WAGSTAFF, Jr., S.S. Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073–1080.MathSciNetzbMATHGoogle Scholar
  89. 1980.
    CHEN, J.R. & PAN, C.D. The exceptional set of Goldbach numbers, I. Sci. Sinica, 23, 1980, 416–430.MathSciNetGoogle Scholar
  90. 1980.
    LIGHT, W.A., FORREST, J., HAMMOND, N., & ROE, S. A note on Goldbach’s conjecture. BIT, 20, 1980, p. 525.MathSciNetzbMATHGoogle Scholar
  91. 1980.
    NEWMAN, D.J. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87, 1980, 693–696.zbMATHGoogle Scholar
  92. 1980.
    PINTZ, J. On Legendre’s prime number formula. Amer. Math. Monthly, 87, 1980, 733–735.MathSciNetzbMATHGoogle Scholar
  93. 1980.
    POMERANCE, C., SELFRIDGE, J.L., & WAGSTAFF, Jr., S.S. The pseudoprimes to 25 · 109. Math. Comp., 35, 1980, 1003–1026.MathSciNetzbMATHGoogle Scholar
  94. 1980.
    VAN DER POORTEN, A.J. & ROTKIEWICZ, A. On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc., A, 29, 1980, 316–321.zbMATHGoogle Scholar
  95. 1981.
    HEATH-BROWN, D.R. Three primes and an almost prime in arithmetic progression. J. London Math. Soc., (2), 23, 1981, 396–414.MathSciNetzbMATHGoogle Scholar
  96. 1981.
    POMERANCE, C. On the distribution of pseudo-primes. Math. Comp., 37, 1981, 587–593.MathSciNetzbMATHGoogle Scholar
  97. 1982.
    POMERANCE, C. A new lower bound for the pseudoprimes counting function. Illinois J. Math., 26, 1982, 4–9.MathSciNetzbMATHGoogle Scholar
  98. 1982.
    WEINTRAUB, S. A prime gap of 682 and a prime arith metic sequence. BIT, 22, 1982, p. 538.MathSciNetzbMATHGoogle Scholar
  99. 1983.
    CHEN, J.R. The exceptional value of Goldbach numbers, II. Sci. Sinica, Ser. A, 26, 1983, 714–731.MathSciNetzbMATHGoogle Scholar
  100. 1983.
    POWELL, B. Problem 6429 (Difference between consecutive primes). Amer. Math. Monthly, 90, 1983, p. 338.Google Scholar
  101. 1983.
    RIESEL, H. & VAUGHAN, R.C. On sums of primes. Arkiv f. Mat., 21, 1983, 45–74.MathSciNetzbMATHGoogle Scholar
  102. 1984.
    DAVIES, R.O. Solution of problem 6429. Amer. Math. Monthly, 91, 1984, p. 64.Google Scholar
  103. 1984.
    IWANIEC, H. & PINTZ, J. Primes in short intervals. Monatsh. Math., 98, 1984, 115–143.MathSciNetzbMATHGoogle Scholar
  104. 1984.
    SCHROEDER, M.R. Number Theory in Science and Communication. Springer-Verlag, New York, 1984.zbMATHGoogle Scholar
  105. 1984.
    WANG, Y. Goldbach Conjecture. World Scientific Publ., Singapore, 1984.zbMATHGoogle Scholar
  106. 1985.
    IVIC, A. The Riemann Zeta-Function. J. Wiley & Sons, New York, 1985.Google Scholar
  107. 1985.
    LAGARIAS, J.C., MILLER, V.S. & ODLYZKO, A.M. Computing7r(x): The Meissel-Lehmer method. Math. Comp., 44, 1985, 537–560.MathSciNetzbMATHGoogle Scholar
  108. 1985.
    LOU, S. & YAO, Q. The upper bound of the difference between consecutive primes. Kexue Tongbao, 8, 1985, 128–129.Google Scholar
  109. 1985.
    POWEL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 58, 1985, p. 46;MathSciNetGoogle Scholar
  110. 1985.
    POWEL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 59 1986, 48–49.Google Scholar
  111. 1985.
    PRITCHARD, P.A. Long arithmetic progressions of primes; some old, some new. Math. Comp., 45, 1985, 263–267.MathSciNetzbMATHGoogle Scholar
  112. 1986.
    BOMBIERI, E., FRIEDLANDER, J.B. & IWANIEC, H. Primes in arithmetic progression to large moduli, I. Acta Math., 156, 1986, 203–251.MathSciNetzbMATHGoogle Scholar
  113. 1986.
    FINN, M.V. & FROHLIGER, J.A. Solution of problem 1207. Math. Mag., 59, 1986, 48–49.MathSciNetGoogle Scholar
  114. 1986.
    MOZZOCHI, C.J. On the difference between consecutive primes. J. Nb. Th., 24, 1986, 181–187.MathSciNetGoogle Scholar
  115. 1986.
    TE RIELE, H.J.J. On the sign of the difference7r(x)–£i(x). Report NM-R8609, Centre for Math. and Comp. Science, Amsterdam, 1986; Math. Comp., 48, 1987, 323–328.zbMATHGoogle Scholar
  116. 1986.
    VAN DE LUNE, J., TE RIELE, H.J.J.,& WINTER, D.T. On the zeros of the Riemann zeta function in the critical strip, IV. Math. Comp., 47, 1986, 667–681.Google Scholar
  117. 1986.
    WAGON, S. Where are the zeros of zeta of s ? Math. Intelligencer, 8, 4, 1986, 57–62.MathSciNetzbMATHGoogle Scholar
  118. 1988.
    ERDÖS, P., KISS, P. & SÂRKÖZY, A. A lower bound for the counting function of Lucas pseudoprimes. Math. Comp., 51, 1988, 315–323.MathSciNetzbMATHGoogle Scholar
  119. 1988.
    PATTERSON, S.J. Introduction to the Theory of the Riemann Zeta-function. Cambridge Univ. Press, Cambridge, 1988.Google Scholar
  120. 1989.
    CONREY, J.B. At least two fifths of the zeros of the Riemann zeta function are on the critical line. Bull. Amer. Math. Soc., 20, 1989, 79–81.MathSciNetzbMATHGoogle Scholar
  121. 1989.
    YOUNG, J. & POTLER, A. First occurrence of prime gaps. Math. Comp., 52, 1989, 221–224.MathSciNetzbMATHGoogle Scholar
  122. 1990.
    JAESCHKE, G. The Carmichael numbers to 1012. Math. Comp., 55, 1990, 383–389.MathSciNetzbMATHGoogle Scholar
  123. 1990.
    PARADY, B.K., SMITH, J.F. & ZARANTONELLO, S. Largest known twin primes. Math. Comp., 55, 1990, 381–382.MathSciNetzbMATHGoogle Scholar

Copyright information

© Paulo Ribenboim 1991

Authors and Affiliations

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

Personalised recommendations