The triangular theorem of eight and representation by quadratic polynomials

Bosma, Wieb; Kane, Ben

doi:10.1090/S0002-9939-2012-11419-4

The triangular theorem of eight and representation by quadratic polynomials
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by Wieb Bosma and Ben Kane PDF

Proc. Amer. Math. Soc. 141 (2013), 1473-1486 Request permission

Abstract:

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n=$ $n(n+1)/2$. In particular, we show that $f(x_1, x_2, \ldots , x_k)=b_1T_{x_1}+\cdots +b_kT_{x_k}$, for fixed positive integers $b_1, b_2, \ldots , b_k$, 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.

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