An estimate for a certain sum extended over the primes of an arithmetical progression

1969 ◽  
pp. 147-164 ◽  
Author(s):  
I. M. Vinogradov
Keyword(s):  
Arithmetical Progression

Minimal invariant spaces in formal topology

1997 ◽  
Vol 62 (3) ◽  
pp. 689-698 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Direct Consequence ◽  
Arithmetical Progression ◽  
Standard Result ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Combinatorial Theorem ◽  
Minimal Property ◽  
Special Instance ◽  
Invariant Spaces ◽  
Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.

scholarly journals Three primes in arithmetical progression

1961 ◽  
Vol 37 (7) ◽  
pp. 329-330
Author(s):  
Saburô Uchiyama
Keyword(s):  
Arithmetical Progression

On prime numbers in an arithmetical progression

1958 ◽  
Vol 4 (1) ◽  
pp. 57-70 ◽  
Author(s):  
Stanisław Knapowski
Keyword(s):  
Prime Numbers ◽  
Arithmetical Progression

scholarly journals W. von Tschirnhaus and algebraic equations with roots in arithmetical progression

1940 ◽  
Vol 6 (3) ◽  
pp. 181-184
Author(s):  
Gino Loria
Keyword(s):  
Algebraic Equation ◽  
Algebraic Equations ◽  
Arithmetical Progression ◽  
German Mathematician ◽  
Ad Libitum

In an interesting appendix to a letter written by John Collins to James Gregory on August 3, 1675, but not published until a few months ago, appear some formulae, given without proof, for expressing the roots of an equation of any degree from the 2nd to the 9th in terms of the coefficients, under the assumption that these roots are in arithmetical progression. The formulae were discovered by the well known contemporary of Leibniz, Baron W. von Tschirnhaus. It is evident that in the case of an equation of degree n this particular assumption imposes n − 2 conditions on the coefficients; so that two of these coefficients can be chosen ad libitum. Tschirnhaus did not go to the trouble of obtaining these relations explicitly, in fact he makes no mention of them, but he gives expressions, in the cases indicated above, for the roots as functions of the first two coefficients of the equation in question, and these coefficients, as we have observed, are arbitrary. It is not known by what approach he arrived at his formulae; it seems likely to us, however, that he expressed the desired roots in terms of two arbitrary unknowns, that he evaluated the sum of these, and the sum of their products two at a time, and that, finally, he equated the results to the first two coefficients of the equation. In this way two equations are obtained, sufficient to determine the two auxiliary unknowns; and the problem can be considered as solved. Without seeming to imply that this procedure was the same as that adopted by the eminent German mathematician, we shall show that by its means one can not only derive his results, but also solve the question in the case of an algebraic equation of any degree.

XXIV.—Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression

1961 ◽  
Vol 65 (4) ◽  
pp. 332-344 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Analogous Problem ◽  
Positive Density ◽  
Arithmetical Progression ◽  
Zero Density

SYNOPSISSets of integers are constructed having the property that n members are in arithmetical progression only if they are all equal; here n is any integer greater than or equal to 3. Previous results have been obtained only for n=3. The problem is generalized in various ways. The analysis can also be applied to construct sets for the analogous problem of geometrical progressions. These sets are of positive density, unlike those of the first kind, which have zero density.

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