Fifteen Theorem

A theorem due to Conway et al. (1997) which states that, if a Positive definite Quadratic Form with integral matrix entries represents all natural numbers up to 15, then it represents all natural numbers. This theorem contains Lagrange's Four-Square Theorem, since every number up to 15 is the sum of at most four Squares.

See also Integer-Matrix Form, Lagrange's Four-Square Theorem, Quadratic Form

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.