III. On the criterion of resolubility in integral numbers of the indeterminate equation f=ax2+a'x2+a“x2+2bx'x”+2b'xx“+2b”x'x=0
Henry J. Stephen Smith · Proceedings of the Royal Society of London · 1864
Abstract It is sufficient to consider the case in which f is an indefinite form of a determinant different from zero. We may also suppose that f is primitive, i. e. that the six numbers a, a', a“, b, b,' b” do not admit of any common divisor. We represent byΩ the greatest common divisor of the minors of the matrix of f, by ΔΩ2 the determinant of f, and by Ωf the contravariant of f, i. e. the form (b2—a'a“)x2+ .....; ΩΔ2 will then be the determinant of F, and Δf its contravariant. By -Ω, -Δ, and -ΩΔ we denote the quotients obtained by dividing Ω, Δ, and Ω, Δ, by the greatest squares contained in them respectively; ω is any uneven prime dividing -Ω, but not -Δ ; δ is any uneven prime dividing -Δ, but not -Ω; and 6 is any uneven prime dividing both 12 and A, and consequently not dividing -ΩΔ. We may then enunciate the theorem— ”The equation f = 0 will or will not be resoluble in integral numbers different from zero according as the equations included in the formulæ (-Ω/δ)=(F/δ), (-Δ/ω)=(f/ω), (--ΩΔ/θ)=(f/θ) (f/θ) are or are not satisfied.“