303,925 research outputs found
Harbingers of Artin's Reciprocity Law. III. Gauss's Lemma and Artin's Transfer
We briefly review Artin's reciprocity law in the classical ideal theoretic
language, and then study connections between Artin's reciprocity law and the
proofs of the quadratic reciprocity law using Gauss's Lemma
Elementary proofs of Berndt's reciprocity laws
Using analytic functional equations, Berndt derived three reciprocity laws connecting five arithmetical sums analogous to Dedekind sums. This paper gives elementary proofs of all three reciprocity laws and obtains them all from a common source, a polynomial reciprocity formula of L. Carlitz
Harbingers of Artin's Reciprocity Law. II. Irreducibility of Cyclotomic Polynomials
In the first article of this series we have presented the history of
auxiliary primes from Legendre's proof of the quadratic reciprocity law up to
Artin's reciprocity law. We have also seen that the proof of Artin's
reciprocity law consists of several steps, the first of which is the
verification of the reciprocity law for cyclotomic extensions. In this article
we will show that this step can be identified with one of Dedekind's proofs of
the irreducibility of the cyclotomic polynomial
Upstream reciprocity in heterogeneous networks
Many mechanisms for the emergence and maintenance of altruistic behavior in
social dilemma situations have been proposed. Indirect reciprocity is one such
mechanism, where other-regarding actions of a player are eventually rewarded by
other players with whom the original player has not interacted. The upstream
reciprocity (also called generalized indirect reciprocity) is a type of
indirect reciprocity and represents the concept that those helped by somebody
will help other unspecified players. In spite of the evidence for the
enhancement of helping behavior by upstream reciprocity in rats and humans,
theoretical support for this mechanism is not strong. In the present study, we
numerically investigate upstream reciprocity in heterogeneous contact networks,
in which the players generally have different number of neighbors. We show that
heterogeneous networks considerably enhance cooperation in a game of upstream
reciprocity. In heterogeneous networks, the most generous strategy, by which a
player helps a neighbor on being helped and in addition initiates helping
behavior, first occupies hubs in a network and then disseminates to other
players. The scenario to achieve enhanced altruism resembles that seen in the
case of the Prisoner's Dilemma game in heterogeneous networks.Comment: 10 figures, Journal of Theoretical Biology, in press (2010
Elliptic Reciprocity
The paper introduces the notions of an elliptic pair, an elliptic cycle and
an elliptic list over a square free positive integer d. These concepts are
related to the notions of amicable pairs of primes and aliquot cycles that were
introduced by Silverman and Stange. Settling a matter left open by Silverman
and Stange it is shown that for d=3 there are elliptic cycles of length 6. For
d not equal to 3 the question of the existence of proper elliptic lists of
length n over d is reduced to the the theory of prime producing quadratic
polynomials. For d=163 a proper elliptic list of length 40 is exhibited. It is
shown that for each d there is an upper bound on the length of a proper
elliptic list over d. The final section of the paper contains heuristic
arguments supporting conjectured asymptotics for the number of elliptic pairs
below integer X. Finally, for d congruent to 3 modulo 8 the existence of
infinitely many anomalous prime numbers is derived from Bunyakowski's
Conjecture for quadratic polynomials.Comment: 17 pages, including one figure and two table
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