Prime numbers contain arbitrarily long arithmetical progressions

Martin Klazar

Pokroky matematiky, fyziky a astronomie (2004)

  • Volume: 49, Issue: 3, page 177-188
  • ISSN: 0032-2423

How to cite

top

Klazar, Martin. "Prvočísla obsahují libovolně dlouhé aritmetické posloupnosti." Pokroky matematiky, fyziky a astronomie 49.3 (2004): 177-188. <http://eudml.org/doc/196625>.

@article{Klazar2004,
author = {Klazar, Martin},
journal = {Pokroky matematiky, fyziky a astronomie},
keywords = {prime number; arithmetic progression; prime number; arithmetic progression},
language = {cze},
number = {3},
pages = {177-188},
publisher = {Jednota českých matematiků a fyziků},
title = {Prvočísla obsahují libovolně dlouhé aritmetické posloupnosti},
url = {http://eudml.org/doc/196625},
volume = {49},
year = {2004},
}

TY - JOUR
AU - Klazar, Martin
TI - Prvočísla obsahují libovolně dlouhé aritmetické posloupnosti
JO - Pokroky matematiky, fyziky a astronomie
PY - 2004
PB - Jednota českých matematiků a fyziků
VL - 49
IS - 3
SP - 177
EP - 188
LA - cze
KW - prime number; arithmetic progression; prime number; arithmetic progression
UR - http://eudml.org/doc/196625
ER -

References

top
  1. Agrawal, M., Kayal, N., Saxena, N., PRIMES is in P, http://www.cse.iitk.ac.in/news/primality.html Zbl1071.11070
  2. Bečvářová, M., Eukleidovy Základy. Jejich vydání a překlady, Prometheus, Praha 2002. (2002) Zbl1024.01030MR1929927
  3. Brun, V., Le crible d’Eratosthène et le théorème de Goldbach, C. R. Acad. Sci. Paris 168 (1919), 544–546. (1919) 
  4. Crandall, R., Pomerance, C., Prime Numbers. A Computational Perspective, Springer-Verlag, New York 2001. (2001) Zbl1088.11001MR1821158
  5. Davis, M., 10.2307/2318447, Amer. Math. Monthly 80 (1973), 233–269. (1973) Zbl0277.02008MR0317916DOI10.2307/2318447
  6. Dirichlet, P. G. L., Beweis des Satzes, daß jede unbegrenzte aritmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abh. Akad. Berlin (1837), 45–71. (1837) 
  7. Edwards, H. M., Riemann’s zeta function, Academic Press, New York-London 1974. (1974) Zbl0315.10035MR0466039
  8. Edwards, H. M., Fermat’s last theorem. A genetic introduction to algebraic number theory, Springer-Verlag, New York 1977. (1977) Zbl0355.12001MR0616635
  9. Erdős, P., 10.1073/pnas.35.7.374, Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 374–384. (1949) Zbl0034.31403MR0029411DOI10.1073/pnas.35.7.374
  10. Friedlander, J., Iwaniec, H., The polynomial X 2 + Y 4 captures its primes, Ann. of Math. (2) 148 (1998), 945–1040. (1998) Zbl0926.11068MR1670065
  11. Friedlander, J., Iwaniec, H., Asymptotic sieve for primes, Ann. of Math. (2) 148 (1998), 1041–1065. (1998) Zbl0926.11067MR1670069
  12. Furstenberg, H., Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977), 204–256. (1977) Zbl0347.28016MR0498471
  13. Furstenberg, H., Katznelson, Y., Ornstein, D., 10.1090/S0273-0979-1982-15052-2, Bull. Amer. Math. Soc. (N. S.) 7 (1982), 527–552. (1982) Zbl0523.28017MR0670131DOI10.1090/S0273-0979-1982-15052-2
  14. Goldstein, L. J., 10.2307/2319162, Amer. Math. Monthly 80 (1973), 599–615. (1973) Zbl0272.10001MR0313171DOI10.2307/2319162
  15. Goldston, D., Yildirim, C. Y., Higher correlations of divisor sums related to primes, I: Triple correlations, Integers 3 (2003), 66 s. (2003) Zbl1118.11039MR1985667
  16. Goldston, D., Yildirim, C. Y., Higher correlations of divisor sums related to primes, III: k -correlations, arXiv:math.NT/0209102, 32 s. Zbl1134.11034
  17. Goldston, D., Yildirim, C. Y., Small gaps between primes, Preprint. 
  18. Gowers, W. T., 10.1007/s00039-001-0332-9, Geom. Funct. Anal. 11 (2001), 465–588. (2001) Zbl1028.11005MR1844079DOI10.1007/s00039-001-0332-9
  19. Gowers, T., Vinogradov’s Three-Primes Theorem, 17 s. http://www.dpmms.cam.ac.uk/~wtg10/ 
  20. Greaves, G., Sieves in number theory, Springer-Verlag, Berlin 2001. (2001) Zbl1003.11044MR1836967
  21. Green, B., Tao, T., The primes contain arbitrarily long arithmetic progressions, arXiv:math.NT/0404188 (verze 1 z 8. dubna 2004), 49 s. (2004) Zbl1191.11025MR2415379
  22. Heath-Brown, D. R., 10.1007/BF02392715, Acta Math. 186 (2001), 1–84. (2001) Zbl1007.11055MR1828372DOI10.1007/BF02392715
  23. Chen, J., 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. (1966) MR0207668
  24. Chen, J., On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176. (1973) MR0434997
  25. Křížek, M., Od Fermatových prvočísel ke geometrii, In: Šolcová, A., Křížek, M., Mink, G., editoři, Matematik Pierre de Fermat. Cahiers du CEFRES č. 28, 131–161. CEFRES, Praha 2002. (2002) 
  26. Křížek, M., Luca, F., Somer, L., 17 lectures on Fermat numbers. From number theory to geometry, Springer-Verlag, New York 2001. (2001) Zbl1010.11002MR1866957
  27. Kučera, L., Kombinatorické algoritmy, SNTL, Praha 1983. (1983) 
  28. Levinson, N., 10.2307/2316361, Amer. Math. Monthly 76 (1969), 225–245. (1969) Zbl0172.06001MR0241372DOI10.2307/2316361
  29. Matijasevič, Ju. V., Diofantovosť perečislimych množestv, Dokl. Akad. Nauk SSSR 191 (1970), 279–282. (1970) 
  30. Matijasevič, Ju. V., Diofantovo predstavlenie množestva prostych čisel, Dokl. Akad. Nauk SSSR 196 (1971), 770–773. (1971) 
  31. Matijasevič, Ju. V., Hilbert’s tenth problem, MIT Press, Cambridge, MA 1993. (1993) 
  32. Nathanson, M. B., Additive Number Theory. The Classical Bases, Springer-Verlag, New York 1996. (1996) Zbl0859.11002MR1395371
  33. Nathanson, M. B., Elementary Methods in Number Theory, Springer-Verlag, New York 2000. (2000) Zbl0953.11002MR1732941
  34. Novák, B., O elementárním důkazu prvočíselné věty, Časopis pro pěstování matematiky 100 (1975), 71–84. (1975) 
  35. Papadimitriou, Ch. H., Computational Complexity, Addison-Wesley, Reading, MA 1994. (1994) Zbl0833.68049MR1251285
  36. Porubský, Š., Fermat a teorie čísel, In: Šolcová, A., Křížek, M., Mink, G., editoři, Matematik Pierre de Fermat. Cahiers du CEFRES č. 28, 49–86. CEFRES, Praha 2002. (2002) 
  37. Pratt, V. R., 10.1137/0204018, SIAM J. Comput. 4 (1975), 214–220. (1975) Zbl0316.68031MR0391574DOI10.1137/0204018
  38. Rabin, M. O., Probabilistic Algorithms., In: J. F. Traub, editor, Algorithms and Complexity, 21–39. Academic Press, New York 1976. (1976) Zbl0384.60001MR0464678
  39. Riemann, B., Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859), 671–680. (1859) 
  40. Rivest, R., Shamir, A., Adleman, L., 10.1145/359340.359342, Comm. ACM 21 1978, 120–126. (1978) Zbl0368.94005MR0700103DOI10.1145/359340.359342
  41. Selberg, A., 10.2307/1969455, Ann. of Math. (2) 50 (1949), 305–313. (1949) Zbl0036.30604MR0029410DOI10.2307/1969455
  42. Serre, J.-P., A Course in Arithmetics, Springer-Verlag, New York 1973. (1973) MR0344216
  43. Shor, P., Algorithms for quantum computation: discrete logarithms and factoring, In: 35th Annual Symposium on Foundations of Computer Science (Santa Fe, NM, 1994), 124–134. IEEE Comput. Soc. Press, Los Alamitos, CA 1994. (1994) Zbl1005.11506MR1489242
  44. Schnirelmann, L., 10.1007/BF01448914, Mat. Ann. 107 (1933), 649–690. (1933) Zbl0006.10402MR1512821DOI10.1007/BF01448914
  45. Stillwell, J., Elements of algebra. Geometry, numbers, equations, Springer-Verlag, New York 1994. (1994) Zbl0832.00001MR1311026
  46. Szemerédi, E., On sets of integers containing no k  elements in arithmetic progression, Acta Arith. 27 (1975), 199–245. (1975) Zbl0335.10054MR0369312
  47. Šnireľman, L. G., Ob additivnych svojstvach čisel, Izvestija donskogo politechničeskogo instituta v Novočerkasske 14 (1930), 3–28. (1930) 
  48. Tao, T., A quantitative ergodic theory proof of Szemerédi’s theorem, arXiv:math.CO/0405251, 51 s. Zbl1127.11011
  49. Tao, T., A quantitative ergodic theory proof of Szemerédi’s theorem (abridged), 20 s. http://www.math.ucla.edu/~tao/preprints/ Zbl1127.11011
  50. Tao, T., A bound for progressions of length  k in the primes, 4 s. http://www.math.ucla.edu/~tao/preprints/ 
  51. Tao, T., A remark on Goldston-Yildirim correlation estimates, 8 s. http://www.math.ucla.edu/~tao/preprints/ 
  52. Tenenbaum, G., Introduction to analytic and probabilistic number theory, Cambridge University Press, Cambridge, U. K. 1995. (1995) Zbl0880.11001MR1342300
  53. Vinogradov, I. M., Predstavlenie něčotnogo čisla summoj trjoch prostych čisel, Dokl. Akad. Nauk SSSR 15 (1937), 291–294. (1937) 
  54. Zagier, D., 10.2307/2975232, Amer. Math. Monthly 104 (1997), 705–708. (1997) Zbl0887.11039MR1476753DOI10.2307/2975232
  55. [unknown], http://www.arxiv.org/ 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.