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G-auss and Legendre had given Post Date: Sat, 26 Jul 2008 8:24:56 +0000 More recently Mertens of Graz has determined the asymptotic values of several numerical functions. Dirichlet gave some attention to prime numbers. G-auss and Legendre had given expressions denoting approximately the asymptotic value of the number of primes inferior to a given limit, but it remained for Bdemann in his memoir, Ueber die Anzakl der Primzahlen unter einer gegebenen Gfrosse, 1859, to give an investigation of the asymptotic frequency of primes which is rigorous. Autor of the post: Undefined Poincare s papers, Sylvester s Post Date: Sat, 26 Jul 2008 8:07:46 +0000 Ap proaching the problem from a different direction, Patnutij Tchebyclieffy formerly professor in the University of St Peters burg (born 1821), established, in a celebrated memoir, Sur les Nombres Premiers, 1850, the existence of limits within which the sum of the logarithms of the primes P, inferior to a given number a, must be comprised. 89 This paper depends on very elementary considerations, and, in that respect, contrasts strongly with Riemann s, which involves abstruse theorems of the integral calculus. Poincare s papers, Sylvester s con traction of Tchebycheff s limits, with reference to the distri bution of primes, and researches of Hadamard (awarded the Grand prix of 1892), are among the latest researches in this line. Autor of the post: Undefined The printing, by the Association Post Date: Sat, 26 Jul 2008 7:47:57 +0000 The enumeration of prime numbers has been undertaken at different times by various mathematicians. In 1877 the British Association began, the preparation of factor-tables, under the direction of G-laisher. The printing, by the Association, of tables for the sixth million marked the completion of tables, to the preparation of which Germany, France, and England contributed, and which enable us to resolve into prime factors every composite number less than 9,000,000. Autor of the post: Undefined He established the theorem Post Date: Sat, 26 Jul 2008 7:28:21 +0000 Miscellaneous contributions to the theory of numbers were made by Cauchy. He showed, for instance, how to find all the infinite solutions of a homogeneous indeterminate equation of the second degree in three variables when one solution is given. He established the theorem that if two congruences, which have the same modulus, admit of a common solution, the modulus is a divisor of their resultant. Autor of the post: Undefined Ternary quadratic forms had been Post Date: Sat, 26 Jul 2008 7:17:17 +0000 Joseph Liouville (1809-1882), professor at the College de France, investigated mainly questions on the theory of quadratic forms of two, and of a greater number of variables. Profound researches were instituted by Ferdinand Gotthold Eisenstein (1823-1852), of Berlin. Ternary quadratic forms had been studied somewhat by Gauss, but the extension from two to three indeterminates was the work of Eisenstein who, in his memoir. Autor of the post: Undefined In inspecting the theory Post Date: Sat, 26 Jul 2008 7:06:38 +0000 Neue Tkeo- reme der liolieren Arithmetik, defined the ordinal and generic characters of ternary quadratic forms of uneven determinant; and, in case of definite forms, assigned the weight of any order or genus. But he did not publish demonstrations of his re sults. In inspecting the theory of binary cubic forms, he was led to the discovery of the first covariant ever considered in analysis. Autor of the post: Undefined Henry John Stephen Smith 90 Post Date: Sat, 26 Jul 2008 6:49:55 +0000 He showed that the series of theorems, relating to the presentation of numbers by sums of squares, ceases when the number of squares surpasses eight. Many of the proofs omitted by Eisenstein were supplied by Henry Smith, who was one of the few Englishmen who devoted themselves to the study of higher arithmetic. Henry John Stephen Smith 90 (1826-1883) was born in Lon don, and educated at Eugby and at Balliol College, Oxford. Autor of the post: Undefined His first paper Post Date: Sat, 26 Jul 2008 6:31:09 +0000 Before 1847 he travelled much in Europe for his health, and at one time attended lectures of Arago in Paris, but after that year he was never absent from Oxford for a single term. In 1861 he was elected Savilian professor of geometry. His first paper on the theory of numbers appeared in 1855. Autor of the post: Undefined They contain much orig inal Post Date: Sat, 26 Jul 2008 6:19:35 +0000 The results of ten years study of everything pub lished on the theory of numbers are contained in his Eeports which appeared in the British Association volumes from 1859 to 1865. These reports are a model of clear and precise exposition and perfection of form. They contain much orig inal matter, but the chief results of his own discoveries were printed in the Philosophical Transactions for 1861 and 1867. Autor of the post: Undefined He contributed also two memoirs Post Date: Sat, 26 Jul 2008 6:08:15 +0000 They treat of linear indeterminate equations and congruences, and of the orders and genera of ternary quadratic forms. He established the principles on which the extension to the gen eral case of n indeterminates of quadratic forms depends. He contributed also two memoirs to the Proceedings of the Royal Society of 1864 and 1868, in the second of which he remarks that the theorems of Jacobi, Eisenstein, and Liou- ville, relating to the representation of numbers by 4, 6, 8 squares, and other simple quadratic forms are dedueible by a uniform method from the principles indicated in his paper. 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