A theorem due to Conway et al. (1997) which states that, if a positive definite quadratic form with integer 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.
Fifteen Theorem
See also
Integer Matrix, Integer-Matrix Form, Lagrange's Four-Square Theorem, Quadratic FormExplore with Wolfram|Alpha
References
Conway, J. H. "Re: The fifteen theorem." math-fun@cs.arizona.edu posting, Sept. 10, 1996.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.Petersen, I. "MathTrek: All Square." Sci. News 169, Mar. 11, 2006. http://www.sciencenews.org/articles/20060311/bob9.asp.Referenced on Wolfram|Alpha
Fifteen TheoremCite this as:
Weisstein, Eric W. "Fifteen Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FifteenTheorem.html