A theorem in finite projective geometry and some applications to number theory

Authors

James Singer

Type: Article

Publication Date: 1938-01-01

Citations: 736

DOI: https://doi.org/10.1090/s0002-9947-1938-1501951-4

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Abstract

A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field of order pn, GF(pn).The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (xi, Xi, x3) and (kxh kx2, kx3) are to be thought of as the same point.The totality of points whose coordinates satisfy the equation uiXi+u2x2+u3x3 = 0, where ui, w2, u3 are marks of the GF(pn), not all zero, is called a line.The plane then consists of p2n+pn+l = q points and q lines; each line contains pn + \ points, j A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer », and there is only one such PG(2, pn) for a given p and » (VB, p. 247, VY, p. 151).Let Ao be a point of a given PG(2, pn), and let C be a collineation of the points of the plane.(A collineation is a 1-1 transformation carrying points into points and lines into lines.)Suppose C carries A o into A\, Ax into * Presented to the Society, October 27, 1934, under a different title;

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Transactions of the American Mathematical Society
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Summary

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The papers comprising this volume explore various fields of number theory, including: the congruent property of solutions of quadratic Diophantine equations, K-groups over the ring of integers of number fields, … The papers comprising this volume explore various fields of number theory, including: the congruent property of solutions of quadratic Diophantine equations, K-groups over the ring of integers of number fields, L-functions of global function fields, automorphic forms, p-adic representations, and more. Applications of number theory, as in design and coding theory, are also explored. The research found in this volume can be seen as a reflection of the influence of Professor Keqin Feng upon the development of number theory studies in China, and this volume is therefore dedicated to him. Number Theory and Related Areas is a valuable reference for experts in these fields, as well as an excellent introduction for graduate students.

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Bibliography

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E. F. Assmus , J. D. Key

Endliche zyklische Ebenen

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Udo Ott

On some power sum problems of Turán and Erdős

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Johan Andersson

B_{h}[g] MODULAR SETS FROM B_{h} MODULAR SETS

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Yadira Caicedo , Jhonny C. Gómez , Carlos A. Trujillo

A set of positive integers A is called a B h [g] set if there are at most g different sums of h elements from A with the same result.This … A set of positive integers A is called a B h [g] set if there are at most g different sums of h elements from A with the same result.This definition has a generalization to abelian groups and the main problem related to this kind of sets, is to find B h [g] maximal sets i.e. those with larger cardinality.We construct B h [g] modular sets from B h modular sets using homomorphisms and analyze the constructions of B h sets by Bose and Chowla, Ruzsa, and Gómez and Trujillo look at for the suitable homomorphism that allows us to preserve the cardinal of this types of sets.

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