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Minkowski gave another geometrical theory of indefinite forms by use of a chain of parallelograms representing a chain of substitutions J. Hence there exists a solution with and only one such solution. Lorch, Archiv Math. Summary in Math. Annalen, 48, , Fricke and F.

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Klein, Automorphen Funktionen, 1, , ; Hurwitz. Physique Hist. Pepin supplemented the investigation by Gauss Arts. An example is summed for all pairs of integers m, n except 0. Let H x, y be a rational function, homogeneous of order 2k in x, y. It will be an invariant of the quadratic form IV if H z, z f is symmetric in z, z'. As a Comptes Rendus Paris, , , , Kristiania, , No. Times, 3, , For corrections and additions, Annales fac.

Toulouse, 3 , 3, , Spiess proved that if A, B, C , A', B', C" have the determinants D, D' and then for all values of a lf a 2 we can determine ft, ft such that identically in t. In order that there shall hold at the same time the similar identity with A and A',. Pepin gave an exposition of the classic theory of binary quadratic forms. Aubry a gave a summary on quadratic forms without exact references. He gave a simple geometrical interpretation by means of the thickest packing of circles.

Dickson obtained necessary and sufficient conditions that two pairs of binary quadratic forms with coefficients in any field or domain of rationality F shall be equivalent under linear transformation with coefficients in F. Define a', c', W by 13 of Gauss. Let a',?

Then Archiv Math. Levy, Paris, , His a,,. I from which we obtain Q' and Q a by replacing a,. By means of this identity we readily obtain all the details of Gauss Arts. These six representations are proper if a is prime to b. If a is a prime, every proper representation of ab is obtained from one proper representation of a and all proper representations of b by means of these product formulas. Uspenskij 1 " 4 applied an algorithm closely analogous to ordinary continued fractions to the reduction of indefinite binary quadratic forms and obtained the periods of reduced forms more rapidly than had Jauss.

Many misprints.

Binary Quadratic Forms with Integer Coefficients — Sage Reference Manual v Quadratic Forms

Petersburg, Cambridge Phil. Use is made also of linear substitutions on x, y with integral coefficients of determinant 1, and consequently of invariants and covariants of the pair of base forms. Chatelet presented Hermite's continual reduction and principal reduced forms from the standpoint of matrices. Any two forms are called equivalent if transformable into each other by linear substitutions with integral coefficients of determinant unity. By use of the continued fraction for a rational number, it is proved that two equivalent reduced forms belong to the same chain.

It is next proved that every form is equivalent to a reduced form. The present method of reduction is said to be essentially that by Mertens and to furnish an introduction to the following paper by Schur. Theiler von algebraischen Zahlen zweiter Ordnung, Diss. Strassburg, Leipzig, , Annalen, 63, , Frobenius called p a Markoff number, studied its properties, and gave explicit expressions for p f q, r in terms of the partial denomi- nators of a continued fraction.

But he did not treat the general Markoff theorem. Bricard proved that every prime 8ql is of the form x 2 2y 2 by a method described in this History, Vol. He gave an explicit analytic expression for R x as an infinite series involving Bessel's function J x.

Another proof was given by G. Van der Corput stated and W. Every divisor of A is representable in the same form A. Primes of certain linear forms are representable by A. Many references are given to sources of various cases of the theorems proved. Humbert proved the theorems stated by Korkine and ZolotarefL G.

This is the analogue of the theorem quoted at the end of the report on Dirichlet 57 on Gauss reduced forms here proved on p. The Hermite 53 reduced forms include principal corresponding to Gauss reduced forms and secondary, and are represented by circles which penetrate' the classic fundamental domain D of the modular group Klein, Smith Humbert proved that, if h ly.

Summary in Comptes Rendus Paris, , , , He found the number of principal Hermite reduced forms which are equivalent to a, b, c , viz. In the main paper only pp. The principal Smith reduced forms are in 1, 1 corre- spondence with the Gauss reduced forms. Humbert gave a theory of reduction much simpler than that of Smith 95 by employing equivalence with respect to the following different subgroup r of the classic modular group.

A positive form a, b, c of proper order i.

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There is one and only one reduced form equivalent in the ordinary sense to a given form. But a positive form of improper order is called reduced modulo 2 if conditions ii hold, there being three reduced forms equivalent in the ordinary sense to a given form. Principal reduced forms are those whose representative circles cut the curved side of D , i.

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There are analogous results for improper orders. For applications, see Humbert of Ch. Thus in a class H of equivalent forms, the reduced form has the minimum coeffi- cients. Let 2 be a set of those forms of H which have the same first coefficient a, and 8 the system of those forms of 2 which are parallel forms their middle coefficients b being congruent modulo a. Amsler applied continued fractions and Farey series Hurwitz to obtain theorems on reduced forms. Gmeiner gave a single process of finding reduced forms whether the determinant is positive or negative.

Mertens gave a simple proof independent of continued fractions of Gauss' theorem that equivalent reduced forms of positive determinant belong to the same period. Two irrational numbers are equivalent under B if and only if their developments into elementary continued fractions coincide after a certain place. A substitution i" Periodic di Mat, 32, , France, 46, , The even and odd substitutions together give all substitutions with integral coefficients of determinants 1 or 1.

There is developed a theory of reduction and equivalence of indefinite binary quadratic forms under the group B of even substitutions. Let N n be the number of solutions of the latter equation. Malo, 22, , , discussed forms of negative determinant capable of representing the same number in several ways.

Metrod, 24, , , reported known results on the representation by ny 2 x 2. Gerardin, 25, , , expressed numbers c 4 1 by x 2 2y 2. In Vol. I of this History references were given for the application of binary quadratic forms to the solution of binomial congruences by Gauss and Legendre, p.

For the applications of binary quadratic forms to factoring with material on congruent forms and idoneal numbers , see pp. A binary quadratic form represents an infinitude of primes, pp.

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  6. Math, de Grece, II, 2, , In his study of biquadratic residues, Gauss this History, Vol. In a posthumous MS. Werke, X 1? Second theorem stated also in a letter to Gauss Feb. He proved Gauss' theorem. For another statement and generalization of Cauchy's results, see the report on Stickel- berger. Berlin, , ; Jour, fur Math. French transl..

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    Ill, XIII. In his proofs, Smith followed the method barely indicated by Jacobi, 6 rather than Cauchy's. Institut de France, 17, , ; Oeuvres, 1 , 3, Critik, , These results were found by induction and not proved. His Table A of factors a n 1 and Table C of primitive roots and exponents are described on p.

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    Bachmann 17 gave an exposition of many of the preceding results, including a proof of Cauchy's 9 theorem for the case in which A is a prime. Cauchy's 11 statement p. Proof repeated by Smith, Coll. Abhandlimg, enthaltend neue Zahlentheoretische Tabellen sammt einer dieselben betreffenden Correspondenz. Jacobi, Progr.