A binomial representation of the 3x + 1 problem

Maurice Margenstern; Yuri Matiyasevich

Acta Arithmetica (1999)

  • Volume: 91, Issue: 4, page 367-378
  • ISSN: 0065-1036

How to cite

top

Maurice Margenstern, and Yuri Matiyasevich. "A binomial representation of the 3x + 1 problem." Acta Arithmetica 91.4 (1999): 367-378. <http://eudml.org/doc/207362>.

@article{MauriceMargenstern1999,
author = {Maurice Margenstern, Yuri Matiyasevich},
journal = {Acta Arithmetica},
keywords = { problem; conjecture; Collatz problem; Ulam's problem; Syracuse algorithm; Kummer's theorem; binomial representation; sums of products of binomial coefficients},
language = {eng},
number = {4},
pages = {367-378},
title = {A binomial representation of the 3x + 1 problem},
url = {http://eudml.org/doc/207362},
volume = {91},
year = {1999},
}

TY - JOUR
AU - Maurice Margenstern
AU - Yuri Matiyasevich
TI - A binomial representation of the 3x + 1 problem
JO - Acta Arithmetica
PY - 1999
VL - 91
IS - 4
SP - 367
EP - 378
LA - eng
KW - problem; conjecture; Collatz problem; Ulam's problem; Syracuse algorithm; Kummer's theorem; binomial representation; sums of products of binomial coefficients
UR - http://eudml.org/doc/207362
ER -

References

top
  1. [1] M. Davis, Yu. Matijasevich and J. Robinson, Hilbert's tenth problem. Diophantine equations: positive aspects of a negative solution, in: Proc. Sympos. Pure Math. 28, Amer. Math. Soc., 1976, 323-378. 
  2. [2] M. Davis, H. Putnam and J. Robinson, The decision problem for exponential Diophantine equations, Ann. of Math. 74 (1961), 425-436. Zbl0111.01003
  3. [3] K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatsh. Math. Phys. 38 (1931), 173-198. Zbl57.0054.02
  4. [4] L. C. Hsu and P. J.-S. Shiue, On a combinatorial expression concerning Fermat's Last Theorem, Adv. in Appl. Math. 18 (1997), 216-219. Zbl0869.11027
  5. [5] E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146. 
  6. [6] J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly 92 (1985), 3-23 (available at: http://www.cecm.sfu.ca/organics/papers/lagarias/index.html). Zbl0566.10007
  7. [7] J. C. Lagarias, The 3x+1 problem and its generalizations, in: J. Borwein et al. (eds.), Organic Mathematics (Burnaby, 1995), Amer. Math. Soc., Providence, RI, 1995. Zbl0566.10007
  8. [8] J. C. Lagarias, 3x+1 Problem Annotated Bibliography, http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/local/anno_bib.ps. 
  9. [9] H. B. Mann and D. Shanks, A necessary and sufficient condition for primality, and its source, J. Combin. Theory Ser. A 13 (1972), 131-134. Zbl0239.10010
  10. [10] M. Margenstern, Frontier between decidability and undecidability: a survey, Theoret. Comput. Sci. (1999), to appear. 
  11. [11] Ju. V. Matijasevič [Yu. V. Matiyasevich], A class of primality criteria formulated in terms of the divisibility of binomial coefficients, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 67 (1977), 167-183, 226-227 (in Russian); English transl.: J. Soviet Math. 16 (1981), 874-885. 
  12. [12] Ju. V. Matijasevič [Yu. V. Matiyasevich], Some arithmetical restatements of the Four Color Conjecture, Theoret. Comput. Sci., to appear; a version is available at: http://logic.pdmi.ras.ru/~yumat/Journal/4cc, mirrored at http://www.informatik.uni-stuttgart.de/ifi/ti/personen/Matiyasevich/Journal/4cc. 
  13. [13] Ju. V. Matijasevič [Yu. V. Matiyasevich], Hilbert's Tenth Problem, Moscow, Fizmatlit, 1993 (in Russian). English transl.: MIT Press, 1993. French transl.: Masson, 1995. See also at: http://logic.pdmi.ras.ru/~yumat/H10Pbook, mirrored at http://www.informatik.uni-stuttgart.de/ifi/ti/personen/Matiyasevich/H10Pbook. 
  14. [14] P. Michel, Busy beaver competition and Collatz-like problems, Arch. Math. Logic 32 (1993), 351-367. Zbl0779.03009
  15. [15] M. Petkovšek, H. S. Wilf and D. Zeilberger, A=B, AK Peters, Wellesley, MA, 1996; http://www.cis.upenn.edu/~wilf/AeqB.html. 
  16. [16] D. Singmaster, Notes on binomial coefficients. I, II, III, J. London Math. Soc. (2) 8 (1974), 545-548, 549-554, 555-560. 
  17. [17] R. Thomas, An update on the Four-Colour Theorem, Notices Amer. Math. Soc. 45 (1998), 848-859 (available at: http://www.ams.org/notices/199807/thomas.ps, http://www.ams.org/notices/199807/thomas.pdf). Zbl0908.05040
  18. [18] H. S. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and 'q') multisum/integral identities, Invent. Math. 108 (1992), 575-633. Zbl0739.05007
  19. [19] G. J. Wirsching, The Dynamical System Generated by the 3n+1 Function, Lecture Notes in Math. 1681, Springer, Berlin, 1998. Zbl0892.11002

NotesEmbed ?

top

You must be logged in to post comments.