Math 2.0

Number theory Definition

Binary quadratic form

Builds on Integral quadratic formIntegersGreatest common divisorSpecial linear group

Definition

A binary quadratic form (with integer coefficients) is a polynomial f(x,y)=ax2+bxy+cy2f(x, y) = ax^2 + bxy + cy^2 in two variables x,yx, y, with coefficients a,b,ca, b, c in the integers. Its discriminant is D=b2−4acD = b^2 - 4ac. The form is primitive if the greatest common divisor of aa, bb, cc is 11. Two forms ff and gg are properly equivalent if g(x,y)=f(αx+βy,γx+δy)g(x, y) = f(\alpha x + \beta y, \gamma x + \delta y) for some matrix (αβγδ)\begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix} in the special linear group SL2(ℤ)\mathrm{SL}_2(\Z).

Remarks

Properly equivalent forms have the same discriminant and represent the same integers. The discriminant always satisfies D≡0D \equiv 0 or 1(mod4)1 \pmod 4. The form is positive definite if and only if D<0D < 0 and a>0a > 0. Some authors write the form as ax2+2bxy+cy2ax^2 + 2bxy + cy^2 and require a,b,ca, b, c to be integers; this is Gauss's convention.

Example: x2+y2x^2 + y^2 has a=1a = 1, b=0b = 0, c=1c = 1, so it is a primitive binary quadratic form of discriminant −4-4.

Non-example: x2+y2+z2x^2 + y^2 + z^2 is a quadratic form with integer coefficients, but it has three variables, so it is not binary.

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General linear group, object, no evidence levelGeneral linear groupobjectGreatest common divisor, concept, no evidence levelGreatest common divis…conceptIdentity matrix, object, no evidence levelIdentity matrixobjectIntegers, object, no evidence levelIntegersobjectIntegral quadratic form, concept, no evidence levelIntegral quadratic fo…conceptMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectBinary quadratic form, definition, no evidence levelBinary quadratic formdefinition
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  1. General linear group object
  2. Greatest common divisor concept
  3. Identity matrix object
  4. Integers object
  5. Integral quadratic form concept
  6. Matrix definition
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  8. Square matrix definition
  9. Linear map definition
  10. Multilinear map definition
  11. Alternating multilinear map definition
  12. Determinant definition
  13. Special linear group object
  14. Binary quadratic form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Binary_quadratic_form
About this page
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Kinddefinition
WritingW0 stub
MSC11E16
DefinesBinary quadratic form
UsesIntegral quadratic form, Integers, Greatest common divisor, Special linear group
Changed2 November 2026

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