The collected mathematical papers of Henry John Stephen Smith: The collected mathematical papers of Henry John Stephen Smith

Glaisher, James Whitbread Lee; Smith, Henry John Stephen

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Multivolume work

Persistent identifier:: 1736276433

Author:: Plato

Title:: Platons Werke

Sub title:: Griechisch und Deutsch mit kritischen und erklärenden Anmerkungen

Year of publication:: 1841

Place of publication:: Leipzig

Identifier (digital):: 1736276433

Language:: Greek, Ancient (to 1453); German

Additional Notes:: Bände 1-26 erschienen 1841-1857

Publisher of the original:: Verlag von Wilhelm Engelmann

Printer:: Engelmann, Wilhelm

Document type:: Multivolume work

Volume

Persistent identifier:: 1737053217

Author:: Plato

Title:: Theätetos

Sub title:: Griechisch und Deutsch mit kritischen und erklärenden Anmerkungen

Scope:: XXVI, 240 Seiten

Year of publication:: 1855

Place of publication:: Leipzig

Identifier (digital):: 1737053217

Signature of the source:: a 1414(20)

Language:: Greek, Ancient (to 1453); German

Publisher of the original:: Verlag von Wilhelm Engelmann

Usage licence:: Public Domain Mark 1.0

Printer:: Engelmann, Wilhelm

Document type:: Volume

Collection:: Philosophy

Section

Title:: [Die in diesem Dialoge vorkommenden Personen sind: Eukleides, Terpsion. Sokrates, Theodoros, Theätetos.]

Document type:: Multivolume work

Structure type:: Section

Section

Title:: 28. [Sokr. Man muss es also untersuchen, [...]]

Document type:: Multivolume work

Structure type:: Section

Art. 23.] REPORT ON THE THEORY OF NUMBERS. 69 cation de l’Algèbre à l’Arithmétique transcendante,’ Crelle, vol. xxix. p. 177. The proof by the same author in vol. xxxv. p. 257, is the same as that in the earlier memoir, only that the properties of the circular functions, which here replace the roots of unity, are in the later memoir deduced immediately from the definition of the sine as the product of an infinite number of factors.) [Addition. From a general theorem of M. Rummer’s (see Arts. 43, 44 of this Report), it appears that the congruence r 2 = ( — 1)^ (A_1) A, mod q, is or is not resoluble, according as gri<*-fl= +1, or = - 1, mod A,—a result which implies the theorem of quadratic reciprocity. This very simple demonstration (which is, however, only a transformation of Gauss’s sixth) appears first to have occurred to M. Liouville (see a note by M. Lebesgue in the Comptes Rendus, vol. li. pp. 12, 13).] 23. Algorithm for the Determination of the Value of the Symbol (pp) — Gauss has shown in the memoir ‘ Demonstrationes et ampliationes novae,’ already quoted, that, if p be a prime number, the value of the symbol ) may be P obtained by developing the vulgar fraction - in a continued fraction, and con- P sidering the evenness or unevenness of a certain function of the quotients and remainders which present themselves in the development. Jacobi has observed (see Crelle, vol. xxx. p. 173) that a much simpler rule may be obtained, by the use of his extension of Legendre’s symbol to the case when p is not a prime. The following is the form in which the rule has been exhibited by Eisenstein (see Crelle, vol. xxvii. p. 319). Let p 1} p 2 be two uneven numbers prime to one another, and let us form by division the series of equations Pi = ^ hip 2 "t ^2_2^3j p 2 = 2lc z p 3 +e 3 p i} Pvl ^ i^fiPfi +14" +1 ’ in which e 2 , e 3 , ... e fl+1 denote positive or negative units, and p 1} p 2 , p$, ••• which are all positive and uneven, form a descending series. Let a- denote the number of the quantities 2 Jc r _ 1 p r + e r p r+1 , in which both p r and e r p r + 1 are of the form 4:ii + 3; then = ( - If. The demonstration of this result flows immediately Jp2 from the definition of Jacobi’s symbol of reciprocity. A numerical example is added (see Disq. Arith,, Art. 328) from which the

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