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Integer-Matrix Form

Let Q(x)=Q(x_1,x_2,...,x_n) be an integer-valued n-ary quadratic form, i.e., a polynomial with integer coefficients which satisfies Q(x)>0 for real x!=0. Then Q(x) can be represented by

Q(x)=x^(T)Ax,

where

A=1/2(partial^2Q(x))/(partialx_ipartialx_j)

is a positive symmetric matrix (Duke 1997). If A has positive entries, then Q(x) is called an integer-matrix form. Conway et al. (1997) have proven that, if a positive integer-matrix quadratic form represents each of 1, 2, 3, 5, 6, 7, 10, 14, and 15, then it represents all positive integers.

See also

Fifteen Theorem

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References

Conway, J. H.; Guy, R. K.; Schneeberger, W. A.; and Sloane, N. J. A. "The Primary Pretenders." Acta Arith. 78, 307-313, 1997.Duke, W. "Some Old Problems and New Results about Quadratic Forms." Not. Amer. Math. Soc. 44, 190-196, 1997.

Referenced on Wolfram|Alpha

Integer-Matrix Form

Cite this as:

Weisstein, Eric W. "Integer-Matrix Form." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Integer-MatrixForm.html

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