188 research outputs found
Nicolaas Govert de Bruijn, the enchanter of friable integers
N.G. de Bruijn carried out fundamental work on integers having only small
prime factors and the Dickman-de Bruijn function that arises on computing the
density of those integers. In this he used his earlier work on linear
functionals and differential-difference equations. We review the relevant work
and also some later improvements by others.Comment: 34 pages, 1 Figur
A top hat for Moser's four mathemagical rabbits
If the equation 1^k+2^k+...+(m-2)^k+(m-1)^k=m^k has an integer solution with
k>1, then m>10^{10^6}. Leo Moser showed this in 1953 by remarkably elementary
methods. His proof rests on four identities he derives separately. It is shown
here that Moser's result can be derived from a von Staudt-Clausen type theorem
(an easy proof of which is also presented here). In this approach the four
identities can be derived uniformly. The mathematical arguments used in the
proofs were already available during the lifetime of Lagrange (1736-1813).Comment: 7 pages. Meanwhile MacMillan and Sondow showed that Lagrange
(1736-1813). can be replaced by Pascal (1623-1662
Integers without large prime factors: from Ramanujan to de Bruijn
A small survey of work done on estimating the number of integers without
large prime factors up to around 1950 is provided. Around 1950 N.G. de Bruijn
published results that dramatically advanced the subject and started a new era
in this topic.Comment: 12 pages, 1 Figur
Counting carefree couples
A pair of natural numbers (a,b) such that a is both squarefree and coprime to
b is called a carefree couple.
A result conjectured by Manfred Schroeder (in his book `Number theory in
science and communication') on carefree couples and a variant of it are
established using standard arguments from elementary analytic number theory.
Also a related conjecture of Schroeder on triples of integers that are pairwise
coprime is proved.Comment: Updated version of 2005 update of 2000 version. Improved and expanded
presentation. In estimate (2) now only a weaker error term than before is
obtaine
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
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