The Triangular Theorem of Eight and Representation by Quadratic Polynomials

We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number is given by Tn = n(n + 1)/2. In particular, we show that f(x1, x2, . . . , xk) = b1Tx1 + · · · + bkTxk , for fixed positive integers b1, b2, . . . , bk, represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if ‘cross-terms’ are allowed in f , we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials.