Weitere Kapitel dieses Buchs durch Wischen aufrufen
When the kind invitation of Ron Graham and Jaroslav Nešetřil, to write in honour of Paul Erdős about aspects of his work, reached us, our first reaction was to follow it with great pleasure. Our second reaction was not as clear: Which one among the many subjects in mathematics, to which he has made fundamental contributions, should we choose?
Finally we just followed the most natural idea to write about an area which just had started to fascinate us: Density Theory for Integer Sequences.
More specifically we add here to the classical theory of primitive sequences and their sets of multiples results for cross-primitive sequences, a concept, which we introduce. We consider both, density properties for finite and infinite sequences. In the course of these investigations we naturally come across the main theorems in the classical theory and the predominance of results due to Paul Erdős becomes apparent. Several times he had exactly proved the theorems we wanted to prove! Many of them belong to his earliest contribution to mathematics in his early twenties.
Quite luckily our random approach led us to the perhaps most formidable period in Erdős’ work. It reminds us about a statement, which K. Reidemeister [18, ch. 8] made about Carl Friedrich Gauss: “…Aber das Epochale ist doch die geniale Entdeckung des Jünglings: die Zahlentheorie.”
Bitte loggen Sie sich ein, um Zugang zu diesem Inhalt zu erhalten
Sie möchten Zugang zu diesem Inhalt erhalten? Dann informieren Sie sich jetzt über unsere Produkte:
F. Behrend, “On sequences of numbers not divisible one by another”, J. London Math. Soc. 10, 42–44, 1935.
P. Erdős, “On primitive abundant numbers”, J. London Math. Soc. 10, 49–58, 1935.
P. Erdős, “Note on sequences of integers no one of which is divisible by any other”, J. Lond. Math. Soc. 10, 136–128, 1935.
P. Erdős, “Generalization of a theorem of Besicovitch”, J. London Math. Soc. 11, 92–98, 1936. CrossRef
H. Davenport and P. Erdős, “On sequences of positive integers”, Acta Arithm. 2,147–151,1937.
S. Pillai, “On numbers which are not multiples of any other in the set”, Proc. Indian Acad. Sci. A 10,392–394, 1939. MathSciNet
P. Erdős, “Integers with exactly k prime factors”, Ann. Math. II 49, 53–66, 1948. CrossRef
P. Erdős, “On the density of some sequences of integers”, Bull. Ann. Math. Soc. 54, 685–692, 1948. CrossRef
N.G. De Bruijn, C. van E. Tengbergen, and D. Kruyswijk, “On the set of divisors of a number”, Nieuw Arch. f. Wisk. Ser II, 23, 191–193, 1949–51.
P. Erdős, Aufgabe 395 in Elem. Math. Basel 16, 21, 1961.
H. Halberstam and K.F. Roth, “Sequences”, Oxford University Press, 1966, Springer Verlag, New York, Heidelberg, Berlin 1983.
R.R. Hall and G. Tenenbaum, “Divisors”, Cambridge Tracts in Mathematics 90, Cambridge University Press, Cambridge, New York, 1988.
P. Erdős, O. Sárközy, and E. Szemerédi, “On a theorem of Behrend”, J. Australian Math. Soc. 7, 9–16, 1967. CrossRef
P. Erdős, O. Sárközy, and E. Szemerédi, “On divisibility properties of sequences of integers”, Coll. Math. Soc. J. Bolyai 2, 35–49, 1970.
R. Ahlswede and L.H. Khachatrian, “On extremal sets without coprimes”, Acta Arithmetica, LXVI 1, 89–99, 1994. MathSciNet
R. Ahlswede and L.H. Khachatrian, “Towards characterising equality in correlation inequalities”, Preprint 93–027, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, to appear in European J. of Combinatorics.
R. Ahlswede and L.H. Khachatrian, “Optimal pairs of incomparable clouds in Multisets”, Preprint 93, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, to appear in Graphs and Combinatorics.
R. Ahlswede and L.H. Khachatrian, “Sharp bounds for cloud-antichains of length two”, Preprint 92–012, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld.
H. Heilbronn, “On an inequality in the elementary theory of numbers”, Cambr. Phil. Soc. 33, 207–209, 1937. CrossRef
H. Rohrbach, “Beweis einer zahlentheoretischen Ungleichung”, J. Reine u. Angew. Math. 177, 193–196, 1937.
- Classical Results on Primitive and Recent Results on Cross-Primitive Sequences
Levan H. Khachatrian
- Springer New York
Neuer Inhalt/© ITandMEDIA, Product Lifecycle Management/© Eisenhans | vege | Fotolia