is a positivesymmetric matrix (Duke 1997). If 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.
be an integer-valued 
-ary quadratic form, i.e.,
a polynomial with integer coefficients
which satisfies 
for real 
. Then 
can be represented by
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.
