The least quadratic non-residue

For a prime number p p , we say a a is a quadratic non-residue modulo p p if there is no integer x x such that x 2 ≡ a mod p x^2\equiv a\bmod {p} . The problem of bounding the least quadratic non-residue modulo p p has a rich mathematical history. Moreover, there have been recent results, especially concerning explicit estimates. In this survey paper we give the history of the problem and explain many of the main achievements, giving explicit versions of these results in most cases. The paper is intended as a self-contained collection of the main ideas that have been used to attack the problem.

Natural Cybernetics and Mathematical History: The Principle of Least Choice in History

The paper follows the track of a previous paper “Natural cybernetics of time” in relation to history in a research of the ways to be mathematized regardless of being a descriptive humanitarian science withal investigating unique events and thus rejecting any repeatability. The pathway of classical experimental science to be mathematized gradually and smoothly by more and more relevant mathematical models seems to be inapplicable. Anyway quantum mechanics suggests another pathway for mathematization; considering the historical reality as dual or “complimentary” to its model. The historical reality by itself can be seen as mathematical if one considers it in Hegel’s manner as a specific interpretation of the totality being in a permanent self-movement due to being just the totality, i.e. by means of the “speculative dialectics” of history, however realized as a theory both mathematical and empirical and thus falsifiable as by logical contradictions within itself as emprical discrepancies to facts. Not less, a Husserlian kind of “historical phenomenology” is possible along with Hegel’s historical dialectics sharing the postulate of the totality (and thus, that of transcendentalism). One would be to suggest the transcendental counterpart: an “eternal”, i.e. atemporal and aspatial history to the usual, descriptive temporal history, and equating the real course of history as with its alternative, actually happened branches of the regions of the world as with only imaginable, counterfactual histories. That universal and transcendental history is properly mathematical by itself, even in a neo-Pythagorean model. It is only represented on the temporal screen of the standard historiography as a discrete series of unique events. An analogy to the readings of the apparatus in quantum mechanics can be useful. Even more, that analogy is considered rigorously and logically as implied by the mathematical transcendental history and sharing with it the same quantity of information as an invariant to all possible alternative or counterfactual histories. One can involve the hypothetical external viewpoint to history (as if outside of history or from “God’s viewpoint to it), to which all alternative or counterfactual histories can be granted as a class of equivalence sharing the same information (i.e. the number choices, but realized in different sequence or adding redundant ones in each branch) being similar and even mathematically isomorphic to Feynman trajectories in quantum mechanics. Particularly, a fundamental law of mathematical history, the law of least choice of the real historical pathway is deducible from the same approach. Its counterpart in physics is the well-known and confirmed law of least action as far as the quantity of action corresponds equivocally to the quantity of information or that of number elementary historical choices.

Der Reiseleiter für Mathematikdidaktik Ervin Deák und das Nichts, das zählt

In the mathematical history course of the mathematical didactic PhD training at the Eötvös Loránd University in Budapest the lecturer Ervin Deák works as a tour guide. A special topic is the development of the number zero, its importance in mathematics and mathematics education. Classification: A30, B50, F30. Keywords: mathematics education, history of mathematics, development of the number zero.