Integer-Matrix Form

Let $Q(x)\equiv Q({\bf x})=Q(x_1, x_2, \ldots, x_n)$ be an integer-valued $n$-ary Quadratic Form, i.e., a Polynomial with integer Coefficients which satisfies $Q(x)>0$ for Real $x\not=0$. Then $Q(x)$ can be represented by

$$Q(x)={\bf x}^{\rm T}{\hbox{\sf A}}{\bf x},$$

where

$${\hbox{\sf A}}={1\over 2}{\partial^2 Q(x)\over\partial x_i\partial x_j}$$

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

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.