Integer-Matrix Form
Let 
be an integer-valued 
-ary Quadratic Form, i.e., aPolynomial 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 matrixform. Conway et al. (1997) have proven that, if a Positive integer matrix quadratic form represents each of1, 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.
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