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.
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References
MEISSEL, E.D.F. Berechnung der Menge von Primzahlen, welche innerhalb der ersten Milliarde natürlicher Zahlen vorkommen. Math. Ann., 25, 1885, 251–257.
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.
VON KOCH, H. Sur la distribution des nombres premiers. Acta Math., 24, 1901, 159–182.
WOLFSKEHL, P. Ueber eine Aufgabe der elementaren Arithmetik. Math. Ann., 54, 1901, 503–504.
LANDAU, E. Handbuch der Lehre von der Verteilung der Primzahlen. Teubner, Leipzig, 1909. Reprinted by Chelsea, Bronx, N.Y., 1974.
LITTLEWOOD, J.E. Sur la distribution des nombres premiers. C.R. Acad. Sci. Paris, 158, 1914, 869–872.
BRUN, V. Le crible d’Eratosthène et le théorème de Goldbach. C.R. Acad. Sci. Paris, 168, 1919, 544–546.
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.
BRUN, V. Le crible d’Erathostène et la théorème de Goldbach. Videnskapsselskapets Skrifter Kristiania, Mat.-nat. Kl. 1920, No. 3, 36 pages.
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.
ERDOS, P. Beweis eines Satzes von Tschebycheff. Acta Sci. Math. Szeged, 5, 1930, 194–198.
HOHEISEL, G. Primzahlprobleme in der Analysis. Sitzungsberichte Berliner Akad. d. Wiss., 1930, 580–588.
SCHNIRELMANN, L. Uber additive Eigenschaften von Zahlen. Ann. Inst. Polytechn. Novocerkask, 14, 1930, 3–28 and Math. Ann., 107, 1933, 649–690.
SKEWES, S. On the differencer(x) — li(x). J. London Math. Soc., 8, 1933, 277–283.
ISHIKAWA, H. Uber die Verteilung der Primzahlen. Sci. Rep. Tokyo Bunrika Daigaku, A, 2, 1934, 27–40.
CRAMÉR, H. On the order of magnitude of the difference between consecutive prime numbers. Acta Arithm., 2, 1937, 2346.
INGHAM, primes. Quart. 255–266.
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.
VAN DER CORPUT, J.G. Sur l’hypothèse de Goldbach pour presque tous les nombres pairs. Acta Arithm., 2, 1937, 266–290.
VINOGRADOV, I.M. Representation of an odd number as the sum of three primes (in Russian). Dokl. Akad. Nauk SSSR, 15, 1937, 169–172.
ESTERMANN, T. Proof that almost all even positive integers are sums of two primes. Proc. London Math. Soc., 44, 1938, 307–314.
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.
ROSSER, J.B. The nth prime is greater than n log n. Proc. London Math. Soc. 45, 1938, 21–44.
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.
VAN DER CORPUT, J.G. Über Summen von Primzahlen und Prim-zahlquadraten. Math. Ann., 116, 1939, 1–50.
LINNIK, Yu.V. On the least prime in an arithmetic progression I. The basic theorem (in Russian). Mat. Sbornik, 15 (57), 1944, 139–178.
BRAUER, A. On the exact number of primes below a given limit. Amer. Math. Monthly, 9, 1946, 521–523.
KHINCHIN, A.Ya. Three Pearls of Number Theory. Original Russian edition in OGIZ, Moscow, 1947. Translation into English published by Graylock Press, Baltimore, 1952.
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.
CLEMENT, P.A. Congruences for sets of primes. Amer. Math. Monthly, 56, 1949, 23–25.
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.
MOSER, L. A theorem on the distribution of primes. Amer. Math. Monthly, 56, 1949, 624–625.
RICHERT, H.E. Über Zerfällungen in ungleiche Primzahlen. Math. Zeits., 52, 1949, 342–343.
SELBERG, A. An elementary proof of the prime number theorem. Annals of Math., 50, 1949, 305–313.
SELBERG, A. An elementary proof of Dirichlet’s theorem about primes in an arithmetic progression. Annals of Math., 50, 1949, 297–304.
SELBERG, A. An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math., 2, 1950, 66–78.
ERDÖS, P. On almost primes. Amer. Math. Monthly, 57, 1950, 404–407.
HASSE, H. Vorlesungen über Zahlentheorie. Springer-Verlag, Berlin, 1950.
SELBERG, A. The general sieve method and its place in prime number theory. Proc. Int. Congr. Math., Cambridge, 1950.
TITCHMARSH, E.C. The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951.
ERDOS, P. On pseudo-primes and Carmichael numbers. Publ. Math. Debrecen, 4, 1956, 201–206.
LEECH, J. Note on the distribution of prime numbers. J. London Math. Soc., 32, 1957, 56–58.
SCHINZEL, A. & SIERPINSKI, W. Sur certaines hypothèses concernant les nombres premiers. Acta Arithm., 4, 1958, 185–208; Erratum, 5, 1959, p. 259.
SCHINZEL, A. Démonstration d’une conséquence de l’hypothèse de Goldbach. Compositio Math., 14, 1959, 74–76.
WRENCH, J.W. Evaluation of Artin’s constant and the twin-prime constant. Math. Comp., 15, 1961, 396–398.
ROSSER, J.B. & SCHOENFELD, L. Approximate formulas for some functions of prime numbers. Illinois J. Math., 6, 1962, 64–94.
AYOUB, R.G. AnIntroduction to the Theory of Numbers. Amer. Math. Soc., Providence, R.I., 1963.
KANOLD, H.J. Elementare Betrachtungen zur Primzahltheorie. Arch. Math., 14, 1963, 147–151.
ROTKIEWICZ, A. Sur les nombres pseudo-premiers de la formeax + b. C.R. Acad. Sci. Paris, 257, 1963, 2601–2604.
WALFISZ, A.Z. Weylsche Exponentialsummen in der neueren Zahfentheorie. VEB Deutscher Verlag d. Wiss., Berlin, 1963.
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.
PAN, C.D. On the least prime in an arithmetic progression. Sci. Record (N.S.), 1, 1957, 311–313.
ROTKIEWICZ, A. Les intervalles contenant les nombres pseudo premiers. Rend. Circ. Mat. Palermo (2), 14, 1965, 278–280.
STEIN, M.L.& STEIN, P.R. New experimental results on the Goldbach conjecture. M.th. Mag., 38, 1965, 72–80.
STEIN, M.L. & STEIN, P.R. Experimental results on additive 2-bases. Math. Comp., 19, 1965, 427–434.
BOMBIERI, E. & DAVENPORT, H. Small differences between prime numbers. Proc. Roy. Soc., A, 293, 1966, 1–18.
LANDER, L.J. & PARKIN, T.R. Consecutive primes in arithmetic progression. Math. Comp., 21, 1967, p. 489.
ROTKIEWICZ, A. On the pseudo-primes of the formax + b. Proc. Cambridge Phil. Soc., 63, 1967, 389–392.
SZYMICZEK, K. On pseudo-primes which are products of distinct primes. Amer. Math. Monthly, 74, 1967, 35–37.
MONTGOMERY, H.L. Zeros of L-functions. Invent. Math., 8, 1969, 346–354.
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.
TITCHMARSH, E.C. The Theory of the Riemann Zeta Function. Clarendon Press, Oxford, 1951.
HUXLEY, M.N. On the difference between consecutive primes. Invent. Math., 15, 1972, 164–170.
HUXLEY, M.N. The Distribution of Prime Numbers. Oxford Univ. Press, Oxford, 1972.
ROTKIEWICZ, A. On a problem of W. Sierpinski. Elem. d. Math., 27, 1972, 83–85.
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;
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.
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.
AYPUB, R.B. Euler and the zeta-function. Amer. Math. Monthly, 81, 1974, 1067–1086.
EDWARDS, H.M. Riemann’s Zeta Function. Academic Press, New York, 1974.
HALBERSTAM, H. & RICHERT, H.E. Sieve Methods. Academic Press, New York, 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.
MAKOWSKI, A. On a problem of Rotkiewicz on pseudoprimes. Elem. d. Math., 29, 1974, p. 13.
MONTGOMERY, H.L. & VAUGHAN, R.C. The exceptional set in Goldbach’s problem. Acta Arithm., 27, 1975, 353–370.
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.
ROSSER, J.B. & SCHOENFELD, L. Sharper bounds for Chebyshev functions0(x) and1(x). Math. Comp., 29, 1975, 243–269.
APOSTOL, T.M. Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976.
BRENT, R.P. Tables concerning irregularities in the distribution of primes and twin primes to 1011. Math. Comp., 30, 1976, p. 379.
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.
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.
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.
WEINTRAUB, S. Seventeen primes in arithmetic progression. Math. Comp., 31, 1977, p. 1030.
BAYS, C.& HUDSON, R.H. On the fluctuations of Littlewood for primes of the form 4n + 1. Math. Comp., 32, 141, 281–286.
HEATH-BROWN, D.R. Almost-primes in arithmetic progressions and short intervals. Math. Proc. Cambridge Phil. Soc., 83, 1978, 357–375.
HEATH-BROWN, D.R. & IWANIEC, H. On the difference between consecutive powers. Bull. Amer. Math. Soc., N.S., 1, 1979, 758–760.
IWANIEC, H. & JUTILA, M. Primes in short intervals. Arkiv f. Mat., 17, 1979, 167–176.
POMERANCE, C. The prime number graph. Math. Comp., 33, 1979, 399–408.
WAGSTAFF, Jr., S.S. Greatest of the least primes in arithmetic progressions having a given modulus. Math. Comp., 33, 1979, 1073–1080.
CHEN, J.R. & PAN, C.D. The exceptional set of Goldbach numbers, I. Sci. Sinica, 23, 1980, 416–430.
LIGHT, W.A., FORREST, J., HAMMOND, N., & ROE, S. A note on Goldbach’s conjecture. BIT, 20, 1980, p. 525.
NEWMAN, D.J. Simple analytic proof of the prime number theorem. Amer. Math. Monthly, 87, 1980, 693–696.
PINTZ, J. On Legendre’s prime number formula. Amer. Math. Monthly, 87, 1980, 733–735.
POMERANCE, C., SELFRIDGE, J.L., & WAGSTAFF, Jr., S.S. The pseudoprimes to 25 · 109. Math. Comp., 35, 1980, 1003–1026.
VAN DER POORTEN, A.J. & ROTKIEWICZ, A. On strong pseudoprimes in arithmetic progressions. J. Austral. Math. Soc., A, 29, 1980, 316–321.
HEATH-BROWN, D.R. Three primes and an almost prime in arithmetic progression. J. London Math. Soc., (2), 23, 1981, 396–414.
POMERANCE, C. On the distribution of pseudo-primes. Math. Comp., 37, 1981, 587–593.
POMERANCE, C. A new lower bound for the pseudoprimes counting function. Illinois J. Math., 26, 1982, 4–9.
WEINTRAUB, S. A prime gap of 682 and a prime arith metic sequence. BIT, 22, 1982, p. 538.
CHEN, J.R. The exceptional value of Goldbach numbers, II. Sci. Sinica, Ser. A, 26, 1983, 714–731.
POWELL, B. Problem 6429 (Difference between consecutive primes). Amer. Math. Monthly, 90, 1983, p. 338.
RIESEL, H. & VAUGHAN, R.C. On sums of primes. Arkiv f. Mat., 21, 1983, 45–74.
DAVIES, R.O. Solution of problem 6429. Amer. Math. Monthly, 91, 1984, p. 64.
IWANIEC, H. & PINTZ, J. Primes in short intervals. Monatsh. Math., 98, 1984, 115–143.
SCHROEDER, M.R. Number Theory in Science and Communication. Springer-Verlag, New York, 1984.
WANG, Y. Goldbach Conjecture. World Scientific Publ., Singapore, 1984.
IVIC, A. The Riemann Zeta-Function. J. Wiley & Sons, New York, 1985.
LAGARIAS, J.C., MILLER, V.S. & ODLYZKO, A.M. Computing7r(x): The Meissel-Lehmer method. Math. Comp., 44, 1985, 537–560.
LOU, S. & YAO, Q. The upper bound of the difference between consecutive primes. Kexue Tongbao, 8, 1985, 128–129.
POWEL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 58, 1985, p. 46;
POWEL, B. Problem 1207 (A generalized weakened Goldbach theorem). Math. Mag., 59 1986, 48–49.
PRITCHARD, P.A. Long arithmetic progressions of primes; some old, some new. Math. Comp., 45, 1985, 263–267.
BOMBIERI, E., FRIEDLANDER, J.B. & IWANIEC, H. Primes in arithmetic progression to large moduli, I. Acta Math., 156, 1986, 203–251.
FINN, M.V. & FROHLIGER, J.A. Solution of problem 1207. Math. Mag., 59, 1986, 48–49.
MOZZOCHI, C.J. On the difference between consecutive primes. J. Nb. Th., 24, 1986, 181–187.
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.
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.
WAGON, S. Where are the zeros of zeta of s ? Math. Intelligencer, 8, 4, 1986, 57–62.
ERDÖS, P., KISS, P. & SÂRKÖZY, A. A lower bound for the counting function of Lucas pseudoprimes. Math. Comp., 51, 1988, 315–323.
PATTERSON, S.J. Introduction to the Theory of the Riemann Zeta-function. Cambridge Univ. Press, Cambridge, 1988.
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.
YOUNG, J. & POTLER, A. First occurrence of prime gaps. Math. Comp., 52, 1989, 221–224.
JAESCHKE, G. The Carmichael numbers to 1012. Math. Comp., 55, 1990, 383–389.
PARADY, B.K., SMITH, J.F. & ZARANTONELLO, S. Largest known twin primes. Math. Comp., 55, 1990, 381–382.
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© 1991 Paulo Ribenboim
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Ribenboim, P. (1991). How Are the Prime Numbers Distributed?. In: The Little Book of Big Primes. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4330-2_5
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