Problems of Number Theory

The preceding chapter contains an exposition of Gauss’ investigations pertaining to binary quadratic forms % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaq6Haamyyai % aadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaamOyaiaa % dIhacaWG5bGaey4kaSIaam4yaiaadMhadaahaaWcbeqaaiaaikdaaa % GccaGGSaaaaa!4276! $$a{x^2} + 2bxy + c{y^2},]$$ , a, b, c ∊ ℤ. Gauss began to study ternary forms % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDhariqtHjhB % LrhDaibaieYlf9irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaceaaq6XaaabCae % aacaWGHbWaaSbaaSqaaiaadMgacaWGRbaabeaaaeaacaWGPbGaaiil % aiaadUgacqGH9aqpcaaIXaaabaGaaG4maaqdcqGHris5aOGaamiEam % aaBaaaleaacaWGPbaabeaakiaadIhadaWgaaWcbaGaam4AaiaacYca % aeqaaaaa!45DC! $$\sum\limits_{i,k = 1}^3 {{a_{ik}}} {x_i}{x_{k,}}]$$ a ik ≡ a ki , in Part V of his Disquisitiones entitled “Digression containing an investigation of ternary forms”. He introduced the notion of a discriminant (he called it a determinant) for such forms and showed that the number of classes of ternary forms with given discriminant is finite. Gauss sketched a program for the further development of a theory of ternary forms, considered their applications to the problem of representation of numbers by means of a sum of three squares, and to the proof of the theorem that every positive integer can be represented as a sum of three triangular numbers or four squares.