Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T17:29:01.510Z Has data issue: false hasContentIssue false
  • English
  • Français

Increasing integer sequences andGoldbach's conjecture

Published online by Cambridge University Press:  20 July 2006

Mauro Torelli
Mauro Torelli*
Affiliation:
Università di Milano, Dipartimento di Scienze dell'Informazione, via Comelico 39, 20135 Milano, Italy; torelli@dsi.unimi.it
Get access

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.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 2: Alberto Bertoni: Climbing summits , April 2006 , pp. 107 - 121
Copyright
© EDP Sciences, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Capell, P. and Narayana, T.V., Knock-out Tournaments, On. Canad. Math. Bull. 13 (1970) 105–109. CrossRef
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.
L. Comtet, Advanced Combinatorics. Reidel, Dordrecht (1974).
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences. Electron. J. Combin. 7 (2000), Research Paper 44, p. 16 (electronic).
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. CrossRef
Dusart, P., The kth prime is greater than k(lnk + lnlnk - 1) for k ≥ 2. Math. Comp. 68 (1999) 411–415. CrossRef
H. Edelsbrunner, Algorithms in Combinatorial Geometry. Springer, Berlin (1987).
Erdös, P. and Lewin, M., d-Complete Sequences of Integers. Math. Comp. 65 (1996) 837–840. CrossRef
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.
R.K. Guy, Unsolved Problems in Number Theory. Springer, New York, 2nd ed. (1994).
D.E. Knuth, The Art of Computer Programming, Vols. 2 and 3. Addison-Wesley, Reading (1997 and 1998).
Lagarias, J.C., The 3x + 1 problem and its generalizations. American Math. Monthly 92 (1985) 3–23. CrossRef
W.J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading (1977).
Melfi, G., Two Conjectures, On about Practical Numbers. J. Number Theory 56 (1996) 205–210. CrossRef
M.B. Nathanson, Additive Number Theory. Springer, New York (1996).
E. Neuwirth, Computing Tournament Sequence Numbers efficiently with Matrix Techniques. Sém. Lothar. Combin. 47 (2002), Article B47h, p. 12 (electronic).
Odlyzko, A.M., Iterated Absolute Values of Differences of Consecutive Primes. Math. Comp. 61 (1993) 373–380. CrossRef
Robbins, D.P., The Story of 1, 2, 7, 42, 429, 7436, ... Math. Intellig. 13 (1991) 12–19. CrossRef
Richstein, J., Verifying the Goldbach Conjecture up to 4 x 1014. Math. Comp. 70 (2001) 1745–1749. CrossRef
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. World-Wide Web URL http://www.research.att.com/~njas/sequences/
N.J.A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences. Academic Press, New York (1995).