Increasing integer sequences and Goldbach's conjecture

Mauro Torelli

RAIRO - Theoretical Informatics and Applications (2006)

  • Volume: 40, Issue: 2, page 107-121
  • ISSN: 0988-3754

Abstract

top
Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.

How to cite

top

Torelli, Mauro. "Increasing integer sequences and Goldbach's conjecture." RAIRO - Theoretical Informatics and Applications 40.2 (2006): 107-121. <http://eudml.org/doc/249725>.

@article{Torelli2006,
abstract = { Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences. },
author = {Torelli, Mauro},
journal = {RAIRO - Theoretical Informatics and Applications},
language = {eng},
month = {7},
number = {2},
pages = {107-121},
publisher = {EDP Sciences},
title = {Increasing integer sequences and Goldbach's conjecture},
url = {http://eudml.org/doc/249725},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Torelli, Mauro
TI - Increasing integer sequences and Goldbach's conjecture
JO - RAIRO - Theoretical Informatics and Applications
DA - 2006/7//
PB - EDP Sciences
VL - 40
IS - 2
SP - 107
EP - 121
AB - Increasing integer sequences include many instances of interesting sequences and combinatorial structures, ranging from tournaments to addition chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements in the sequence. The paper introduces and compares several of these classes of sequences, discussing recurrence relations, enumerative problems and questions concerning shortest sequences.
LA - eng
UR - http://eudml.org/doc/249725
ER -

References

top
  1. P. Capell and T.V. Narayana, On Knock-out Tournaments. Canad. Math. Bull.13 (1970) 105–109.  
  2. L. Carlitz, D.P. Roselle and R.A. Scoville, Some Remarks on Ballot-Type Sequences of Positive Integers. J. Combinatorial Theory, Ser. A 11 (1971) 258–271.  
  3. L. Comtet, Advanced Combinatorics. Reidel, Dordrecht (1974).  
  4. M. Cook and M. Kleber, Tournament sequences and Meeussen sequences. Electron. J. Combin.7 (2000), Research Paper 44, p. 16 (electronic).  
  5. J-M. Deshouillers, H.J.J. te Riele and Y. Saouter, New experimental results concerning the Goldbach conjecture, in Proc. 3rd Int. Symp. on Algorithmic Number Theory. Lect. Notes Comput. Sci.1423 (1998) 204–215.  
  6. P. Dusart, The kth prime is greater than k(lnk + lnlnk - 1) for k ≥ 2. Math. Comp.68 (1999) 411–415.  
  7. H. Edelsbrunner, Algorithms in Combinatorial Geometry. Springer, Berlin (1987).  
  8. P. Erdös and M. Lewin, d-Complete Sequences of Integers. Math. Comp.65 (1996) 837–840.  
  9. P.C. Fishburn and F.S. Roberts, Uniqueness in finite measurement, in Applications of Combinatorics and Graph Theory to the Biological and Social Sciences, edited by F.S. Roberts. Springer, New York (1989) 103–137.  
  10. R.K. Guy, Unsolved Problems in Number Theory. Springer, New York, 2nd ed. (1994).  
  11. D.E. Knuth, The Art of Computer Programming, Vols. 2 and 3. Addison-Wesley, Reading (1997 and 1998).  
  12. J.C. Lagarias, The 3x + 1 problem and its generalizations. American Math. Monthly92 (1985) 3–23.  
  13. W.J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading (1977).  
  14. G. Melfi, On Two Conjectures about Practical Numbers. J. Number Theory56 (1996) 205–210.  
  15. M.B. Nathanson, Additive Number Theory. Springer, New York (1996).  
  16. E. Neuwirth, Computing Tournament Sequence Numbers efficiently with Matrix Techniques. Sém. Lothar. Combin.47 (2002), Article B47h, p. 12 (electronic).  
  17. A.M. Odlyzko, Iterated Absolute Values of Differences of Consecutive Primes. Math. Comp.61 (1993) 373–380.  
  18. D.P. Robbins, The Story of 1, 2, 7, 42, 429, 7436, ... Math. Intellig.13 (1991) 12–19.  
  19. J. Richstein, Verifying the Goldbach Conjecture up to 4 x 1014. Math. Comp.70 (2001) 1745–1749.  
  20. N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. World-Wide Web URL  URIhttp://www.research.att.com/~njas/sequences/
  21. N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences. Academic Press, New York (1995).  

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.