@article{RM_1987_42_5_a18,
author = {P. Ribenboim},
title = {Prime number records (a~new chapter for the {Guinness} book of records)},
journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
year = {1987},
volume = {42},
number = {5},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/RM_1987_42_5_a18/}
}
P. Ribenboim. Prime number records (a new chapter for the Guinness book of records). Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 42 (1987) no. 5. http://geodesic.mathdoc.fr/item/RM_1987_42_5_a18/
[1] Dirchlet G. L., “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält”, Abh. d. Königl. Acad. d. Wiss., 1837, 45–81, Reprinted in Werke, Vol. 1, 315–350, G. Reimer, Berlin, 1889
[2] Eisenstein F. G., “Aufgaben”, J. f. d. reine u. angew. Math., 27 (1844), 87; Reprinted in Mathematice Werke, Vol. 1, Chelsea Publ. Co., New York, 1975, p. 112
[3] Bouniakowsky V., “Nouveaux théorèmes relatifs à la distiction des nombres premiers et à la décomposition des entiers en facteurs”, Mem. Acad. Sci. St. Petersbourg, (6), Sci. Math. Phys., 6 (1857), 305–329
[4] Lucas E., “Sur las recherche des grands nombres premiers”, Assoc. Francáise p. l'A-vanc des Sciences, 5 (1876), 61–68
[5] Pepin T., “Sur la formule $2^{2^n}+1$”, C. R. Acad. Sci. Paris, 85 (1877), 329–331
[6] Meissel E. D. F., “Berechnung der Menge von Primzahlen, welche innerhalb des ersten Milliarde naturlichen Zahlen vorkommen”, Math. Annalen, 25 (1885), 251–257 | DOI | MR
[7] Wendt E., “Elementarer Beweis des Satzess dass in jeder unbegrenzter arithmetischen Progression $my+1$ unendlich viele Primzahlen vorkommen”, J. f. d. reine u. angew. Math., 115 (1885), 85–88
[8] Thue A., “Mindre meddelelser II. Et bevis for at primtallenes antal er uendeligh”, Archiv f. Math. og Naturv. Kristiania, 19:4 (1897), 3–5; Reprinted in Selected Mathematical papers of A. Thue, Universitetsforlaget, Oslo, 1977 | MR | Zbl
[9] Chrystal G., Algebra, Vol. II, A. C. Black, London, 1900, 2nd edition; Reprinted by Chelsea, New York, 1964 | Zbl
[10] Torelli G., “Julia totalità dei numeri primi fino a un limite assegnato”, Atti. d. Reale Acad. d. Sci. Fis. e Mat. di Napoli, (2), 11 (1902), 1–222
[11] Bachmann P., Niedere Zahlentheorie, Vol. II, Teubner, Leipzig, 1902, Reprinted by Chelsea, New York, 1968 | Zbl
[12] Landau E., “Neuer Beweis des Primzahlsatzes und Beweis des Primidealsatzes”, Math. Ann., 56 (1903), 645–670 | DOI | MR | Zbl
[13] Cipolla M., “Sui numeri compositi $p$, che verificano la congruenza di Fermat $a^{P-1}\equiv1\pmod P$”, Annali di Matematica, (3), 9 (1904), 139–160 | DOI
[14] Dickson L. E., “A new extension of Dirichlet's theorem on prime numbers”, Messenger of Math., 33 (1904), 155–161
[15] Landau E., Handbuch der Lehre der Verteilung der Primzahlen, Teubner, Leipzig, 1909, Reprinted by Chelsea, New York, 1974
[16] Lehmer D. N., List of prime numbers from 1 to 10 006 721, Reprinted by New York, Hafner, 1956
[17] Lehmer D. N., Factor table for the first ten millions containing the smallest factor of every number not divisible by 2, 3, 5, or 7 between the limits 0 and 10 017 000, Reprinted by Hafner, 1956
[18] Carmichael R. D., “On composite numbers $P$ which satisfy the Fermat congruence $a^{P-1}\equiv1\pmod P$”, Amer. Math. Monthly., 19 (1912), 22–27 | DOI | MR
[19] Litllewood J. E., “Sur la distribution des nombres premiers”, C. R. Acad. Sci. Paris., 158 (1914), 1869–1872
[20] Weil H., “Über die Gleichverteilung von Zahlen mod Eins”, Math. Ann., 77 (1916), 313–352 | DOI | MR
[21] Brun V., “Le crible d'Eratosthène et le théorème de Goldbach”, C. R. Acad. Sci. Paris, 168 (1919), 544–546 | Zbl
[22] Brun V., “La serie $\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+ \frac{1}{19}+\frac{1}{29}+\frac{1}{31}+\frac{1}{41}+\frac{1}{43}+ \frac{1}{59}+\frac{1}{61}+\dots$ où les dénominateurs sont “nombres premiers jumeaux” est convergente ou finie”, Bull. Sci. Math. (2), 43 (1919), 100–104 ; 124–128 | Zbl
[23] Brun V., “Le crible d'Eratosthène et le théorème de Goldbach”, Videnskapsselskapets Skrifter Kristiania, Mat.- nat. Kl., 1920, no. 3
[24] Carmichael R. D., “Note on Euler'sc $\varphi$-function”, Bull. Amer. Math. Soc., 28 (1922), 109–110 | DOI | Zbl
[25] 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 | DOI | MR
[26] Polya G., Szogö G., Aufgaben und Lehrsätze aus der Analysis, 2 vols., (4rh edition, 1970), Springer-Verlag, Berlin, 1924; Poia G., Segë G., Zadachi i teoremy analiza, Nauka, M., 1978
[27] Lehmer D. H., “An extended theory of Lucas's functions”, Annals Math., 31 (1930), 419–448 | DOI | MR | Zbl
[28] Hoheisel G., “Primzahlprobleme in der Analysis”, Berliner Sitzungsberichte, 1930, 580–588 | Zbl
[29] Schnirelmann L., “Über additive Eigenschaften von Jahlen”, Math., 107 (1932), 649–690 | MR
[30] Lehmer D. H., “On Euler's totient function”, Bull. Amer. Math. Soc., 33 (1932), 745–757 | DOI | MR
[31] Landau E., “Über den Wienerschen neuen Weg zum Primidealsatz”, Sitzungober, Berliner Acad. d. Wiss, 1932, 514–521 | Zbl
[32] Ikewes I., “On the difference $\pi(x)-\operatorname{Li}(x)$”, J. London Math. Soc, 8 (1933), 277–283 | DOI
[33] Ishikawa H., “Über die Verteilung der Primzahlen”, Sci. Rep. Tokyo Bunrika Daigaku, A, 1934, no. 2, 27–40 | MR | Zbl
[34] Erdös P., “On the difference of consecutive primes”, Quart. J. Oxford, 1935, no. 6, 124–128 | DOI | MR | Zbl
[35] Lehmer D. H., “On Lucas's test for the primality of Mersenne numbers”, J. London Math. Soc., 10 (1935), 162–165 | Zbl
[36] Cramer H., “On the order of magnitude of the difference between consecutive prime numbers”, Acta Arithm., 2 (1937), 23–46
[37] Ingham A. E., “On the difference between consecutive primes”, Quart. J. Oxford, 1937, no. 8, 255–266 | DOI | Zbl
[38] Landau E., Über einige neuere Fortschritte der additiven Zahlentheorie, Cambridge Univ. Press, Cambridge, 1937, Reprinted by Stechert–Hafner, New York, 1964 | Zbl
[39] Van der Corput J. G., “Sur l'hypothèse de Goldbach pour presque tous les nombres pairs”, Acta Arithm., 1937, no. 2, 266–290 | MR | Zbl
[40] Vinogradov I. M., “Predstavlenie nechetnogo chisla summoi trekh prostykh chisel”, DAN SSSR, 15 (1937), 169–172
[41] Vinogradov I. M., “Some theorems concerning the theory of primes”, Mat. Sbornik, N. S., 1937, no. 2(44), 179–195
[42] Hall M., “Divisors of second order sequences”, Bull. Amer. Math. Soc., 43 (1937), 78–80 | DOI | Zbl
[43] Bilharz H., “Primdivisoren mit vorgegebener Primitivwurzel”, Mat. Ann., 114 (1937), 476–492 | DOI | MR | Zbl
[44] Ricci G., “Su la congettura di Goldbach e la constante de Schnirelmann”, Annali. Scuola Norm. Sup. Pisa, 1937, no. 6(2), 71–116 | MR | Zbl
[45] Hardy G. H., Wright E. M., An introduction to the theory of Numbers, (4-th edition, 1959), Clarendon Press, Oxford, 1938 | MR | Zbl
[46] Estermann T., “Proof that almost all even positive integers are sums of two primes”, Proc. London Math. Soc., 44 (1938), 307–314 | DOI | Zbl
[47] Hardy G. H., Heilbronn H., “Edmund Landau”, J. London Math. Soc., 13 (1938), 302–310 | DOI | MR | Zbl
[48] Poulet P., “Tables des nombres composés vérifiant le théorème de Fermat pour le module 2, jusqu' à 100.000.000”, Sphinx, 1938, no. 8, 52 ; Corrections: Math. Comp., 25 (1971), 944–945; 26 (1972), 814 | Zbl | MR
[49] Rankin R. A., “The difference between consecutive prime numbers”, J. London Math. Soc., 13 (1938), 242–247 | DOI | Zbl
[50] Rosser B., “The $n$-th prime is greater than $n\log n$”, Proc London Math. Soc., 45 (1938), 21–44 | DOI | Zbl
[51] Chudakov N. G., “O plotnosti sovokupnosti chetnykh chisel, nepredstavimykh kak summa dvukh nechetnykh prostykh”, Izv. AN SSSR. Ser. mat., 1938, no. 1, 25–40 | Zbl
[52] Chernick J., “On Fermat's simple theorem”, Bull. Amer. Math. Soc., 45 (1939), 269–274 | DOI | Zbl
[53] Krasner M., “À propos du critère de Sophie Germain–Furthwängler pour le premier cas du thèorème de Fermat”, Mathematica Cluj., 16 (1940), 109–114 | MR | Zbl
[54] Reiner I., “Functions not formulas for primes”, Amer. Math. Monthly, 50 (1943), 619–621 | DOI | MR
[55] Chowla S., “There exists an infinity of 3-combinations of primes in A. P.”, Proc. Lahore Phil. Soc., 6 (1944), 15–16 | MR | Zbl
[56] Linnik Yu. V., “On the least prime in an arithmetic progression, I. The basic theorem”, Mat. sb., 15(57) (1944), 139–178 | MR | Zbl
[57] Buck R. C., “Prime-representing functions”, Amer. Math. Monthly, 53 (1946), 265 | DOI | MR | Zbl
[58] Mills W. H., “A prime-representing function”, Bull. Amer. Math. Soc., 53 (1947), 604 | DOI | MR | Zbl
[59] Xinchin A. Ya., Tri zhemchuzhiny teorii chisel, Nauka, M., 1979 | MR
[60] Renui A., “On the representation of even numbers as the sum of a prime and an almost prime”, DAN SSSR, 56 (1947), 455–458 | MR
[61] Klee V. L., “On a conjecture of Carmichael”, Bull. Amer. Math. Soc., 53 (1947), 1183–1187 | DOI | MR
[62] Vinogradov I. M., “Ob otsenke trigonometricheskikh summ s prostymi chislami”, Izv. AN SSSR. Ser. mat., 12:3 (1948), 225–248 | Zbl
[63] Liénard R., Tables fondamentles à 50 décimales des sommes $S_n$, $u_n$, $\Sigma_n$, Centre de Docum. Univ., Paris, 1948 | MR
[64] Weil A., Sur les Courbes Algebriques et les Variétés que s'en Deduisent, Hermann, Paris, 1948 | MR
[65] Erdös P., “On the converse of Fermat's theorem”, Bull. Amer. Math. Soc., 45 (1949), 269–274
[66] Clement P. A., “Congruences for sets of primes”, Amer. Math. Monthly, 56 (1949), 23–25 | DOI | MR | Zbl
[67] Erdös P., “On a new method in the elementary number theory which leads to an elementary proof of the prime number theorem”, Proc. Nat. Acad. Sci. USA, 35 (1949), 374–384 | DOI | MR | Zbl
[68] Richert H. E., “Über Zerflällungen in ungleiche Primzahlen”, Math. Z., 52 (1949), 342–343 | DOI | MR | Zbl
[69] Selberg A., “An elementary proof of the prime number theorem”, Annals. Math., 50 (1949), 305–313 | DOI | MR | Zbl
[70] Selberg A., “An elementary proof of Dirichlet's theorem about primes in an arithmetic progression”, Annals Math., 50 (1949), 297–304 | DOI | MR | Zbl
[71] Giuga G., “Su una presumibile propriety caratteristica dei numeri primi. 1-st Lombardo Sci. Lett. Rend. C1”, Sci. Mat. Nat. (3), 14(83) (1950), 511–528 | MR
[72] Erdös P., “On almost primes”, Amer. Math. Monthly, 57 (1950), 404–407 | DOI | MR | Zbl
[73] Hasse H., Vorlesungen über Zahlentheorie, Springer-Verlag, Berlin, 1950 | MR | Zbl
[74] Selberg A., “The general sieve method and its place in prime number theory”, Proc. Int. Congr., Cambridge, Mas., 1950, no. 1, 286–292 | MR
[75] Beeger N. G. W. H., “On even numbers dividing $2^m-2$”, Amer. Math. Monthly, 58 (1951), 553–555 | DOI | MR | Zbl
[76] Wright E. M., “A prime-representing function”, Amer. Math. Monthly, 58 (1951), 616–618 | DOI | MR | Zbl
[77] Nagell T., Introduction to Number Theory, Almqvist, Stockholm, 1951, Reprinted by Chelsea, New York, 1964 | MR | Zbl
[78] Titchmarsh E. C., The theory of the Riemann zeta-function, Press, Clarendon Oxford, 1951 ; Titchmarsh E., Teoriya dzeta-funktsii Rimana, IL, M., 1953 | MR | Zbl
[79] Dénes P., “An extension of Legendre's criterion in connection with the first case of Fermat's last theorem”, Publ. Math. Debrecen, 1951, no. 2, 115–120 | MR | Zbl
[80] Davenport H., The higher arithmetic, Hutchison, London, 1952 ; Davenport G., Vysshaya arifmetika, Nauka, M., 1965 | Zbl | Zbl
[81] Sierpinski W., “Sur une formule donnant tous les nombres premiers”, C. R. Acad. Sci. Paris., 235 (1952), 1078–1079 | MR | Zbl
[82] Erdös P., Mirsky L., “The distribution of values of the divisor function $d(n)$”, Proc. London Math. Soc. (3), 1952, no. 2, 257–271 | DOI | MR | Zbl
[83] Trost E., Primzahlen, Birkhäuser, Basel, 1953 ; Trost E., Prostye chisla, Fizmatgiz, M., 1959 | MR | Zbl
[84] Pyatetskii–Shapiro I. I., “O raspredelenii prostykh chisel v posledovatelnostyakh vida $\bigl[f(n)\bigr]$”, Mat. sb., 33:3 (1953), 559–566
[85] Goldberg K., “A table of Wilson quotient and the third Wilson prime”, J. London Math. Soc., 28 (1953), 252–256 | DOI | MR | Zbl
[86] Robinson R. M., “Mersenne and Fermat numbers”, Proc. Amer. Math. Soc., 1954, no. 5, 842–846 | DOI | MR | Zbl
[87] Ward M., “Prime divisors of second order recurring sequence”, Duke Math. J., 21 (1954), 607–614 | DOI | MR
[88] Kuhn P., “Über die Primteiler eines Polynoms”, Proc. Intern. Congress Math., no. 2, Amsterdam, 1954, 35–37
[89] Skewes S., “On the difference $\pi(x)-\operatorname{Li}(x)$, II”, Proc. London Math. Soc., 1955, no. 5, 48–70 | DOI | MR | Zbl
[90] Erdös P., “On pseudo-primes and Carmichael numbers”, Publ. Math. Debrecen, 1956, no. 4, 201–206 | MR | Zbl
[91] Prachar K., Primzahlverteilung, (2nd edition 1978), Springer-Verlag, Berlin, 1957 ; Prakhar K., Raspredelenie prostykh chisel, Mir, M., 1967 | MR | Zbl | MR
[92] Robinson R. M., “The converse of Fermat's theorem”, Amer. Math. Monthly, 64 (1957), 703–710 | DOI | MR | Zbl
[93] Robinson R. M., “A report on primes of the form $k2^n+1$ and on factors of Fermat numbers”, Proc. Amer. Math. Soc., 9 (1958), 673–681 | DOI | MR | Zbl
[94] Sierpinski W., “Sur les nombres premiers de la forme $n^n+1$”, L'Enseign. Math. (2), 4 (1958), 211–212 | MR | Zbl
[95] Baker G. L. Gruenberger E. J., Primes in the thousandth million, The Rand Corp., Sante Monica, 1958
[96] Erdös P., “Some remarks on Euler's $\varphi$-function”, Acta Arithm., 4 (1958), 10–19 | MR | Zbl
[97] Schinzel A., Sierpinski W., “Sur certaines hypothèses concernant les nombres premiers. Remarque”, Acta Arithm., 4 (1958), 185–208 ; 5 (1959), 259 | MR | Zbl | MR
[98] Baker C. L., Gruenberger E. J., The first six million prime numbers, Microcard Found, Medison, 1959
[99] Lehmer D. H., “On the exact number of primes less than a given limit”, Illinois J. Math., 1959, no. 3, 381–388 | MR | Zbl
[100] Sierpinski W., “Sur les nombres premiers ayant des chiffres initiaux et finals donnés”, Acta Arithm., 5 (1959), 265–266 | MR | Zbl
[101] Inkeri K., “Tests for primality”, Annales Acad. Sci. Fennicae, Ser. A, 1, 279, Helsinki, 1960 | MR | Zbl
[102] Sierpinski W., “Sur un problème concernant les nombres $k\cdot2^n+1$”, Elem. d. Math., 15 (1960), 73–74 | MR | Zbl
[103] Rotkiewicz A., “Démonstration arithmétique d'existence d'une infinité de nombres premiers de la forme $nk+1$”, L'Enseign. Math. (2), 1962, no. 7, 277–280 | MR | Zbl
[104] Wrench J. W., “Evaluation of Artin's constant and the twin-prime constant”, Math. Comp., 15 (1961), 396–398 | DOI | MR | Zbl
[105] Aigner A., “Folgen der Art $ar^n+b$ welche nur teilbare Zahlen Liefern”, Math. Nachr., 23 (1961), 259–264 | Zbl
[106] Ward M., “The prime divisors of Fibonacci numbers”, Pacific. J. Math., 11 (1961), 259–264 | MR
[107] Schinzel A., “Remarks on the paper “Sur certaines hypotheses concernant les nombres premiers””, Acta Arithm., 1961, no. 7, 1–8 | MR | Zbl
[108] Shanks D., Solved and unsolved problems in number theory, Spartan, Washington, 1962, Reprinted by Chelsea, New York, 1978 | MR
[109] Crocker R., “A theorem on pseudo-primes”, Amer. Math. Monthly, 69 (1962), 540 | DOI | MR | Zbl
[110] Cohn H., Advanced Number Theory, Wiley, New York, 1962, Reprinted by Dover, New York, 1980 | MR | Zbl
[111] Rosser B., Schoenfeld L., “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 1962, no. 6, 64–94 | MR | Zbl
[112] Bateman P. T., Horn R. A., “A heuristic asymptotic formula concerning the distribution of prime numbers”, Math. Comp., 16 (1962), 363–367 | DOI | MR | Zbl
[113] Ayoub R. G., An introduction to the theory of numbers, Amer. Math. Soc., Providence, R. I., 1963 | MR | Zbl
[114] Estermann T., “Note on a paper of Rotkiewicz”, Acta. Arithm., 1963, no. 8, 465–467 | MR | Zbl
[115] Kanold H. J., “Elementare Betrachtungen zur Primzahltheorie”, Arch. Math., 14 (1963), 147–151 | DOI | MR | Zbl
[116] Neubauer G., “Eine empirische Untersuchung zur Mertenschen Funktiony”, Numer. Math., 5 (1963), 1–13 | DOI | MR | Zbl
[117] Rankin R. A., “The difference between consecutive prime numbers, V”, Proc. Edinburgh Math. Soc. (2), 13 (1963), 331–332 | DOI | MR | Zbl
[118] Rotkiewicz A., “Sur les nombres pseudo-preimers de la forme $ax+b$”, C. R. Acad. Sci. Paris., 257 (1963), 2601–2604 | MR | Zbl
[119] Selfridge J. L., “Solution to problem 4995 (proposed by O. Ore)”, Amer. Math. Monthly, 70 (1963), 101 | DOI | MR
[120] Vorobev N. N., Chisla Fibonachchi, Nauka, M., 1984
[121] Edwards A. W. F., “Infinite coprime sequences”, Math. Gazette, 48 (1964), 416–422 | DOI | MR | Zbl
[122] Willians C. P., “On formulae for the $n$-th prime”, Math. Gaz., 48 (1964), 413–415 | DOI | MR
[123] Kapferer H., “Verifizierung des symmetrischen Teils der Fermatschen Vermutung für unendlich viele paarweise teilerfremde Exponenten E”, Journal f. d. reine u. angew. Math., 214:5 (1964), 360–372 | MR | Zbl
[124] Rohrbach H., Weis J., “Zum finiten Fall des Bertrandschen Postulats”, Journ. f. d. reine u. angew. Math., 214:5 (1964), 432–440 | MR | Zbl
[125] Sierpinski W., Elementary Theory of Numbers, Hafner, New York, 1964 | MR
[126] Shen M. K., “On checking the Goldbach conjecture”, Nordisk Tidskr., 1964, no. 4, 243–245 | MR | Zbl
[127] Graham R. L., “A Fibonacci-like sequences of composite numbers”, Math. Mag., 37 (1964), 322–324 | Zbl
[128] Gilles D. B., “Three new Mersenne primes and a statistical theory”, Math. Comp., 18 (1964), 93–98 | DOI | MR
[129] Siegel C. L., “Zu zwei Bemerkungen Kummers”, Nachr. Akad. d. Wiss. Göttingen Math. Phys. Kl., II, 1964, 51–62 | MR
[130] Rotkiewicz A., “Sur les nombres de Mersenne dépourvus de facteurs carres et sur les nombres naturels $n$ tells que $n^2/2^n-2$”, Matem. Vesnik (Beograd), 1965, no. 2(17), 78–80 | MR | Zbl
[131] Stein M. L., Stein P. R., “New experimental results on the Goldbach conjecture”, Math. Mag., 38 (1965), 72–80 | MR | Zbl
[132] Kloss K. E., “Some number theoretical calculations”, J. Res. Nat. Bureau of Stand. B., 69 (1965), 335–336 | MR | Zbl
[133] Bateman P. T., Horn R. A., “Primes represented by irreducible polynomials in one variable”, Proc. Symp. Pure Math., Theory of Numbers, VIII, Amer. Math. Soc., 1965, 119–132 | MR
[134] 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”, Kexue Tongbao, 17 (1966), 385–386 | MR
[135] Bombieri E., Davenport H., “Small diffences between prime numbers”, Proc. Roy. Soc. A, 293 (1966), 1–18 | DOI | MR | Zbl
[136] Lehman R. S., “On the difference $\pi(x)-\operatorname{Li}(x)$”, Acta Arithm., 1966, no. 11, 397–410 | MR | Zbl
[137] Hasse H., “Über die Dichte der Primzahlen $p$, fur die eine vogegebene ganzrationale Zahl $a\pm0$ von gerader bzw. ungerader Ordnung $\mod p$ ist”, Mat. Annalen., 168 (1966), 19–23 | DOI | MR
[138] Kruyswijk D., On the congruence $u^{p-1}\equiv1\pmod{p^2}$, (in Dutch), Amsterdam, Math. Centrum, 1966 | Zbl
[139] Grosswald E., Topics from the Theory of Numbers, Macmillan, New York, 1966 | MR | Zbl
[141] Goodstein R. L., Wormell C. P., “Formulae for primes”, Math. Gaz., 51 (1967), 35–38 | DOI | MR
[142] Stark H. M., “A complete determination of the complex quadratic fields of classnumber one”, Michigan Math. J., 14 (1967), 1–27 | DOI | MR | Zbl
[143] Jones M. F., Lal M., Blundon W. J., “Statistics on certain large primes”, Math. Comp., 21 (1967), 103–107 | DOI | MR | Zbl
[144] Kolesnik G. A., “O raspredelenii prostykh chisel v posledovatelnostyakh vida $[n^c]$”, Mat. zametki, 2 (1967), 117–128 | MR | Zbl
[145] Lander L. J., Parkin T. R., “Consecutive primes in arithmetic progression”, Math. Comp., 21 (1967), 489 | DOI | Zbl
[146] Scymiczek K., “On pseudo-primes which are products of distinct primes”, Amer. Math. Monthly, 74 (1967), 35–37 | DOI | MR
[147] Hooley C., “On Artin's conjecture”, Journal f. d. reine u. angew. Math., 225 (1967), 209–220 | MR | Zbl
[148] Dudley U., “History of a formula for primes”, Amer. Math. Monthly, 76 (1969), 23–28 | DOI | MR | Zbl
[149] Montgomery H. L., “Zeros of $L$-functions”, Invent. Math., 1969, no. 8, 346–354 | DOI | MR | Zbl
[150] Rosser J. B., Yohe J. M., Schoenfeld L., “Rigorous computation of the zeros of Riemann zeta-function”, Inform. Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), I, North-Holland, Amsterdam, 1969, 70–76 | MR
[151] Baker A., “On the class number of imaginary quadratic fields”, Bull. Amer. Math. Soc., 77 (1971), 678–684 | DOI | MR | Zbl
[152] Gandhi J. M., “Formulae for the $n$-th prime”, Proc. Washington State Univ. Conf. on Number Theory, Wash. St. Univ., Pullman, Wash., 1971, 96–106 | MR | Zbl
[153] Montgomery H. L., “Topics in Multiplicative Number Theory”, Lect. Notes in Math., no. 227, Springer-Verlag, 1971 ; Montgomeri, Multiplikativnaya teoriya chisel, Mir, M., 1974 | MR | Zbl | MR
[154] Sergushov S. A., “O probleme bliznetsov”, Uch. zapiski Yaroslavskogo gos. ped. in-ta, 82 (1971), 85–86
[155] Brillhart J., Tonascia J., Weinberger P., “On the Fermat quotient”, Computers in Number Theory, Academic Press, New York, 1971, 213–222 | MR
[156] Uchida K., “Glass numbers of imaginary abelian number fields, III”, Tôhoku Math. J., 23 (1971), 573–580 | DOI | MR | Zbl
[157] Van den Eynden G., “A proof of Gandhi's formula for the $n$-th prime”, Amer. Math. Monthly, 79 (1972), 625 | DOI | MR | Zbl
[158] Deshouillers J. M., “Nombres premiers de la forme $[n^c]$”, C. R. Acad. Sci. Paris, Sér A, 282 (1972), 131–133 | MR
[159] Huxley M. N., “On the difference between consecutive primes”, Invent. Math., 15 (1972), 164–170 | DOI | MR | Zbl
[160] Rotkiewicz A., “On a problem of W. Sierpinski”, Elem. d. Math., 27 (1972), 83–85 | MR | Zbl
[161] Iwasowa K., Lectures on $p$-adic $L$-functions, Annals of Math. Studies, Princeton Univ. Press, Princeton, 1972 | MR
[162] Ribenboim P., Algebraic Numbers, Wiley-Interscience, New York, 1972 | MR | Zbl
[163] Davis M., “Hilbert's tenth problem is unsolvable”, Amer. Math. Monthly., 80 (1973), 233–269 | DOI | MR | Zbl
[164] Karst E., “New quadratic forms with high density of primes”, Elem. d. Math., 28 (1973), 116–118 | MR | Zbl
[165] Apostol T. M., “Another elementary proof of Euler's formula for $\zeta(2n)$”, Ater. Math. Monthly., 80 (1973), 425–431 | DOI | MR | Zbl
[166] Chen J. R., “On the representation of a large even integer as a sum of a prime and the product of at most two primes”, Sci. Sinica, 16 (1973), 157–176 ; 21 (1978), 421–430 | MR | Zbl | MR | Zbl
[167] Montgomery H. L., “The pair correlation of zeros of the zeta-function”, Analytic Number Theory (Proc. Symp. Pure Math., Vol. XXIV, St. Louis, 1972), Amer. Math. Soc., Providence, RI, 1973, 181–193 | MR
[168] Vaughan R. C., “A remark on divisor function $d(n)$”, Glasgow Math. J., 14 (1973), 54–55 | DOI | MR | Zbl
[169] Rademacher H., Topics in Analytic Number Theory, Springer-Verlag, New York, 1975 | MR
[170] Halberstam H., Richert H. E., Sieve Methods, Academic Press, New York, 1974 | MR | Zbl
[171] Golomb S. W., “A direct interpretation of Gandhi's formula”, Amer. Math. Monthly, 81 (1974), 752–754 | DOI | MR | Zbl
[172] Ayoub R., “Euler and the zeta-function”, Amer. Math. Monthly, 81 (1974), 1067–1086 | DOI | MR | Zbl
[173] Brent R. P., “The distribution of small gaps between successive primes”, Math. Comp., 28 (1974), 315–324 | DOI | MR | Zbl
[174] Edwards H. M., Riemann's zeta-function, Academic Press, New York, 1974 | Zbl
[175] Levinson N., “More than one third of zeros of Riemann's zeta-function are on $\sigma=1/2$”, Adv. in Math., 13 (1974), 383–436 | DOI | MR | Zbl
[176] Makowski A., “On a problem of Rotkiewicz on pseudo-primes”, Elem. d. Math., 29 (1974), 13 | MR
[177] Shanks D., Wrench J. W., “Brun's constant”, Math. Comp., 28 (1974), 293–299 | DOI | MR | Zbl
[178] Angell I. O., Godwin H. J., “Some factorizations of $10^n\pm1$”, Math. Comp., 28 (1974), 307–308 | DOI | MR | Zbl
[179] Borucki L. J., Diaz J. B., “A note on primes, with arbitrary initial or terminal decimal ciphers in Diriclet arithmetic progressions”, Amer. Math. Monthly, 81 (1974), 1001–1002 | DOI | MR | Zbl
[180] Pomerance C., “On Carmichael's conjecture”, Proc. Amer. Math. Soc., 43 (1974), 297–298 | DOI | MR | Zbl
[181] Ellison J. E., Mendes–France M., Les Nombres Premiers, Hermann, Paris, 1975 | MR | Zbl
[182] Stöwener F., “Simultanbeweis des Fermatschen und Wilsonschen Satzes”, Elem. d. Math., 30 (1975), 39–40 | MR
[183] Ernvall R., “A formula for the least prime greater than a given integer”, Elem. d. Math., 30 (1975), 13–14 | MR | Zbl
[184] Montgomery H. L., Vaughan R. C., “The exponentional set in Goldbach's problem”, Acta Arithm., 27 (1975), 353–370 | MR | Zbl
[185] Ross P. M., “On Chen's theorem that each large even number has the form $p_1+p_2$ or $p_1+p_2p_3$”, J. London Math. Soc. (2), 10 (1975), 500–506 | DOI | MR | Zbl
[186] Rosser J. B., Schoenfeld L., “Sharper bound, for Chebyshev functions $\theta(x)$ and $\psi(x)$”, Math. Comp., 29 (1975), 243–269 | DOI | MR | Zbl
[187] Swift J. D., “Table of Carmichael numbers to $10^9$”, Math. Comp., 29 (1975), 338–339 | DOI
[188] Metsänkylä T., “On the cyclotomic invariants of Iwasawa”, Math. Scand., 37 (1975), 61–75 | MR | Zbl
[189] Adams W. W., Goldstein L. J., Introduction to number theory, Prentice Hall, Englewood Cliffs, New York, 1976 | MR | Zbl
[190] Lehmer D. H., “Strong Carmichael numbers”, J. Austral. Math. Soc. A, 21 (1976), 508–510 | DOI | MR | Zbl
[191] Rabin M. O., Probabilistic algorithms in symposium in new directions and recent results in algorithms and complexity, 21–31, New York, Academic Press, 1976 | MR
[192] Jones J. P., Sato D., Wada H., Wiens D., “Diophantine representation of the set of prime numbers”, Amer. Math. Monthly, 83 (1976), 449–464 | DOI | MR | Zbl
[193] Apostol T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976 | MR
[194] Niven I., Powell B., “Primes in certain arithmetic progressions”, Amer. Math. Monthly, 83 (1976), 467–469 | DOI | MR | Zbl
[195] Schoenfeld L., “Sharper bounds for Chebyshev functions $\theta(x)$ and $\psi(x)$, II”, Math. Comp., 30 (1976), 337–360 | DOI | MR | Zbl
[196] Vaughan R. C., “A note on Schnirelmann's approach to Goldbach's problem”, Bull. London Math. Soc., 8 (1976), 245–250 | DOI | MR | Zbl
[197] Mendelsohn N. S., “The equation $\psi(x)=k$”, Math. Mag., 49 (1976), 37–39 | MR | Zbl
[198] Malm D. E. G., “On Monte-Carlo primality tests”, Notices Amer. Math. Soc., 24 (1977), A-529, abstract 77T -A22
[199] Solovay R., Strassen V., “A fast Monte-Carlo test for primality”, SIAM J. Comput., 1977, no. 6, 84–85 | DOI | MR | Zbl
[200] Deshouillers J. M., Sur la constante de Schnirelmann, Sem. Delange–Pisot–Poitou, 17-e année, 1975/6, fasc. 2, Paris, 1977, exp. No G16 | MR
[201] Huxley M., “Small differences between consecutive primes II”, Mathematika, 24 (1977), 142–152 | MR
[202] Jutila M., “On Linnik's constant”, Math. Scand., 41 (1977), 45–62 | MR | Zbl
[203] Powell B., “Proof of a special case of Dirichlet's theorem”, Fibonacci Quart., 15 (1977), 167–169 | MR | Zbl
[204] Weintraub S., “Seventeen primes in arithmetic progression”, Math. Comp., 31 (1977), 1030 | DOI | MR | Zbl
[205] Zagier D., “The first 50 million prime numbers”, Math. Intelligencer, 1977, 7–19 | DOI | Zbl
[206] Johnson W., “On the non-vanishing of Fermat's quotient $\pmod p$”, Journal f.d. reine u. angew. Math., 292 (1977), 196–200 | MR | Zbl
[207] Pomerance C., “On composite $n$ for which $\varphi(n)\mid n-1$, II”, Pacific J. Math., 69 (1977), 177–186 | MR
[208] Powell B., DeLeon M. J., “Problem E2631 (and its solution)”, Amer. Math. Monthly, 84 (1977), 57 ; 85 (1978), 279–280 | DOI | MR | DOI | MR
[209] Rivest R. L., Shamir A., Adleman L., “A method for obtaining digital signatures and public-key cryptosystems”, Comm. ACM, 21 (1978), 120–126 | DOI | MR | Zbl
[210] Yorinaga M., “Numerical computation of Carmichael numbers”, Math. J. Okayama Univ., 20 (1978), 151–163 | MR | Zbl
[211] Ellison W., Ellison F., Théorie des Nombres. Abrégé d'Histoire des Mathematiques, Vol. I, Chapter V, § VI, ed. J. Dieudonné, Hermann, Paris, 1978 | MR
[212] Metsänkylä T., “Iwasawa invariants and Kummer congruences”, J. Nb. Th., 10 (1978), 510–522 | DOI | MR | Zbl
[213] Wagstaff S. S., “The irregular primes to 125 000”, Math. Comp., 32 (1978), 583–591 | DOI | MR | Zbl
[214] Williams H. C., “Some primes with interesting digit patterns”, Math. Comp., 32 (1978), 1306–1310 | DOI | MR | Zbl
[215] Iwaniec H., “Almost-primes represented by quadratic polynomials”, Invent Math., 47 (1978), 171–188 | DOI | MR | Zbl
[216] Slowinski D., “Searching for the 27-th Mersenne prime”, J. Recr. Math., 11 (1978/79), 258–261 | MR
[217] Hill J. R., “Carmichael numbers with three prime factors”, Notices Amer. Math. Sco., 26 (1979), A-374
[218] Atkin A. O. L., Rickert N. W., “On a larger pair of twin-primes. Abstract 79T-A132”, Notices Amer. Math. Soc., 26 (1979), A-373
[219] Chen J. R., “On the least prime in arithmetic progression and theorems concerning the zeros of Dirichlet's $L$-functions, II”, Sci. Sinica, 22 (1979), 859–889 | MR | Zbl
[220] Grosswald E., Hagis P., “Arithmetic progressions consisting only of primes”, Math. Comp., 33 (1979), 1343–1352 | DOI | MR | Zbl
[221] Heath-Brown D. R., Iwaniec H., “On the difference between consecutive powers”, Bull. Amer. Math. Soc. N. J., 1979, no. 1, 758–760 | DOI | MR | Zbl
[222] Iwaniec H., Jutila M., “Primes in short intervals”, Arkiv. f. Mat., 17 (1979), 167–179 | DOI | MR
[223] Pomerance C., “The prime number graph”, Math. Comp., 33 (1979), 399–408 | DOI | MR | Zbl
[224] Rotkiewicz A., Wasen R., “On a number-theoretical series”, Publ. Math. Debrecen, 26 (1979), 1–4 | MR | Zbl
[225] Bayer P., “Sobre el indice de irregularidad de los numeros primos”, Collect. Math., 30 (1979), 11–20 | MR | Zbl
[226] Ferrero B., Washington L. C., “The Iwasawa invariant $\mu_p$ vanishes for abelian fields”, Ann. Math., 109 (1979), 377–395 | DOI | MR | Zbl
[227] Ribenboim P., 13 Lectures on Fermat's last theorem, Springer-Verlag, New York, 1979 | MR | Zbl
[228] Williams H. C., Seah E., “Some primes of the form $(a^n-1)/(a-1)$”, Math. Comp., 33 (1979), 1337–1342 | DOI | MR | Zbl
[229] Washington L. C., The infinitude of primes via commutative algebra, 1980
[230] Pomerance C., Selfridge J. L., Wagstaff S. S., “The pseudo-primes to $25\cdot10^9$”, Math. Comp., 36 (1980), 1003–1026 | DOI | MR
[231] Cohen G. L., Hagis P., “On the number of prime factors of $n$ if $\varphi(n)/(n-1)$”, Nieuw. Arch. Wisk. (3), 1980, 177–185 | MR | Zbl
[232] Wagstaff S. S., “Large Carmichael numbers”, Math. J. Okayama Univ., 22 (1980), 33–41 | MR | Zbl
[233] Brent R. P., “The first occurrence of certain large prime gaps”, Math. Comp., 35 (1980), 1435–1436 | DOI | MR | Zbl
[234] Erdös P., Straus E. G., “Remarks on the difference between consecutive primes”, Elem. d. Math., 35 (1980), 115–118 | MR | Zbl
[235] Light W. A., Forrest J., Hammond N., Roe S., “A note on Goldbach's conjecture”, BIT, 20 (1980), 525 | DOI | MR | Zbl
[236] Newman D. J., “Simple analytic proof of the prime number theorem”, Amer. Math. Monthly, 87 (1980), 693–696 | DOI | MR | Zbl
[237] Pintz J., “On Legendre's prime number formula”, Amer. Math. Monthly, 87 (1980), 733–735 | DOI | MR | Zbl
[238] Van den Eynden C., “Proofs that $\sum1/P$ diverges”, Amer. Math. Monthly, 87 (1980), 394–397 | DOI | MR | Zbl
[239] Van der Poorten A. J., Rotkiewiz A., “On strong pseudo-primes in arithmetic progressions”, J. Austral. Math. Soc. A, 29 (1980), 316–321 | DOI | MR | Zbl
[240] Newman M., Shanks D., Williams H. C., “Simple groups of square order and an interesting sequence of primes”, Acta Arithm., 38 (1980), 129–140 | MR | Zbl
[241] Powell B., “Primitive densities of certain sets of primes”, J. Nb. Th., 12 (1980), 210–217 | DOI | MR | Zbl
[242] Guy R. K., Unsolved problems in number theory, Springer-Verlag, New York, 1981 | MR
[243] Dixon J. D., “Asymptotically fast factorization of integers”, Math. Comp., 36 (1981), 255–260 | DOI | MR | Zbl
[244] Konheim A. G., Cryptography: A Primer, Wiley-Interscience, New York, 1981 | MR | Zbl
[245] Pomerance C., “Recent developments in primality testingy”, Math. Intelligencer, 3:3 (1981), 97–105 | DOI | MR
[246] Heath-Brown D. R., “Three primes and an almost prime in arithmetic progression”, J. London Math. Soc. (2), 23 (1981), 396–414 | DOI | MR | Zbl
[247] Leavitt W. G., Mullin A. A., “Primes differing by a fixed integer”, Math. Comp., 37 (1981), 581–585 | DOI | MR | Zbl
[248] Lehmer D. H., “On Fermat's quotient, base two”, Math. Comp., 36 (1981), 289–290 | DOI | MR | Zbl
[249] Pomerance C., “On the distribution of pseudo-primes”, Math. Comp., 37 (1981), 587–593 | DOI | MR | Zbl
[250] Weintraub S., “A large prime gap”, Math. Comp., 36 (1981), 279 | DOI | MR | Zbl
[251] Hua L. K., Introduction to number theory, Springer-Verlag, Berlin, 1982 | MR
[252] Buhler J. P., Crandall R. F., Penk M. A., “Primes of the form $n!+1$ and $2\cdot3\cdot5\dots p\pm1$”, Math. of Comp., 38 (1982), 639–643 | DOI | MR | Zbl
[253] Williams H. C., “The influence of computers in the development of number theory”, Comp. and Moths. with Appl., 1982, no. 8, 75–93 | MR | Zbl
[254] Jones J. P., “Universal diophantine equation”, J. Symb. Logic., 47 (1982), 549–571 | DOI | MR | Zbl
[255] Diamond H. G., “Elementary methods in the study of the distribution of prime numbers”, Bull. Amer. Math. Soc., 7 (1982), 533–589 | DOI | MR
[256] Pomerance C., “A new lower bound for the pseudo-primes counting function”, Illinois J. Math., 26 (1982), 4–9 | MR | Zbl
[257] Pritchard P. A., “18 primes in arithmetic progression”, J. Recr. Math., 15 (1982/3), 288 | MR
[258] Powell B., “Problem E2956”, Amer. Math. Monthly, 89 (1982), 498 | DOI | MR
[259] Yates S., Repunits and Repetends, Star Publ. Co., Boynton Beach, Florida, 1982 | MR
[260] Schinzel A., Selected Topic on Polynomials, Univ. of Michigan Press, Ann. Arbor, 1982 | MR
[261] Adleman L. M., Pomerance C., Rumely R. S., “On distinguishing prime numbers from composite numbers”, Annals Math. (2), 117 (1983), 173–206 | DOI | MR | Zbl
[262] Brillhart J., Lehmer D. H., Selfridge J. L., Tuckerman B., Wagstaff S. S., “Factorization of $b^n\pm1$, $b=2,3,5,6,7,10,11,12$ up to high powers”, Contemporary Math., 22, Amer. Math. Soc., Providence, R.I., 1983 | MR
[263] Keller W., “Factors of Fermat numbers and large primes of the form $k\cdot2^n+1$”, Math. Comp., 41 (1983), 661–673 | DOI | MR | Zbl
[264] Rumely R. S., “Recent advances in primality testing”, Notices Amer. Math. Soc., 30:5 (1983), 475–477 | MR | Zbl
[265] Conrey J. B., “Zeros of derivatives of Riemann's $xi$-function on the critical line”, J. Nb. Th., 16 (1983), 49–74 | DOI | MR | Zbl
[266] Keller W., “Large twin prime pairs related to Mersenne numbers”, Abstracts Amer. Math. Soc., 4 (1983), 490
[267] Powell B., “Problem 6429”, Amer. Math. Monthly, 90 (1983), 338 | MR
[268] Riesel H., Vaughan R. C., “On sums of primes”, Arkirf. Mat., 21 (1983), 45–74 | DOI | MR | Zbl
[269] Jaeschke G., “On the smallest $k$ such that $k\cdot2^N+1$ are composite”, Math. Comp., 40 (1983), 381–384 | DOI | MR | Zbl
[270] Yates S., “Titanic primes”, J. Recr. Math., 16 (1983/4), 250–260 | MR
[271] Ribenboim P., “1093”, Math. Intelligencer, 5 (1983), 28–34 | DOI | MR | Zbl
[272] Schroeder M. R., “Where is the next Mersenne prime hiding?”, Math. Intelligencer, 5:3 (1983), 31–33 | DOI | MR | Zbl
[273] Wagstaff S. S., “Divisors of Mersenne numbers”, Math. Comp., 40 (1983), 385–397 | DOI | MR | Zbl
[274] Cohen H., Lenstra H. W., “Primality testing and Jacobi sums”, Math. Comp., 42 (1984), 297–330 | DOI | MR | Zbl
[275] Dixon J. D., “Factorization and primality tests”, Amer. Math. Monthly, 91 (1984), 333–352 | DOI | MR | Zbl
[276] Nicolas J. L., “Tests de primalité”, Expo. Math., 2 (1984), 223–234 | MR | Zbl
[277] Pomerance C., “Lecture Notes on Primality Testing and Factoring (Notes by Gagola Jr.)”, Math. Assoc. America, 1984, Notes No 4
[278] Williams H. C., “Factoring on a computer”, Math. Intelligencer, 6:3 (1984), 29–36 | DOI | MR | Zbl
[279] Daboussi H., “Sur le théorèm des nombres premiers”, C. R. Acad. Sci. Paris, (1), 298 (1984), 161–164 | MR | Zbl
[280] Iwaniec H., Pintz J., “Primes in short intervals”, Monatsh. Math., 98 (1984), 115–143 | DOI | MR | Zbl
[281] Schröder M. R., Number theory in science and communication, Springer-Verlag, New York, 1984
[282] Wang Y., Goldbach conjecture, World Scientific Publ., Singapore, 1984 | MR | Zbl
[283] Keller W., “The 17-th prime of the form $5\cdot2^n+1$”, Abstract Amer. Math. Soc., 6:1 (1985), 121
[284] Riesel H., Prime numbers and computer methods for factorization, Birkhäuser, Boston, 1985 | MR | Zbl
[285] Adleman L. M., Heath-Brown D. R., “The first case of Fermat's last theorem”, Invent. Math., 79 (1985), 409–416 | DOI | MR | Zbl
[286] Fouvry E., “Theoreme de Brun–Titchmarsh; application au théorème de Fermat”, Invent. Math., 79 (1985), 383–407 | DOI | MR | Zbl
[287] Lagarias J. C., “The set of primes dividing the Lucas numbers has density 2/3”, Pacific J. Math., 118 (1985), 19–23 | MR
[288] Ribenboim P., “An extention of Sophie Germain's method to a wide class of diophantine equations”, Journal f.d. reine u. angew. Math., 356 (1985), 49–66 | MR | Zbl
[289] Balasubramanian R., Conrey J. B., Heath-Brown D. R., “Asymptotic mean square of the product of the Riemann zeta-function and a Dirichelet polynomial”, Journal f.d. reine u. angew. Math., 357 (1985), 161–181 | MR | Zbl
[290] Ivich A., The Riemann zeta-function, J. Wiley Sons, New York, 1985 | MR
[291] Lagaias J., Miller V. S., Odlyzko A., Computing $\pi(x)$: The Meissel–Lehmer method, 1985
[292] Maier H., “Smalt differences between prime numbers”, Michigan Math. J., 32 (1985), 221–225 | DOI | MR | Zbl
[293] Odlyzko A., te Riele H. J. J., “Disproof of Mertens conjecture”, Journal f.d. reine u. angew. Math., 357 (1985), 138–160 | MR | Zbl
[294] Pintz J., “On primes in short intervals, II”, Stud. Sci. Math. Hung., 1985 | MR
[295] Pritchard P. A., “Long arithmetic progressions of primes; some old, some new”, Math. Comp., 45 (1985), 263–267 | DOI | MR | Zbl
[296] 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., Centrum v. Wisk. e. Inform. Amsterdam, 1985 (preprint)