## Integer-Matrix Form

Let be an integer-valued -ary Quadratic Form, i.e., a Polynomial with integer Coefficients which satisfies for Real . Then can be represented by

where

is a Positive symmetric matrix (Duke 1997). If A has Positive entries, then 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.

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.