Universal mixed sums of generalized $4$- and $8$-gonal numbers

An integer of the form $P_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers ${\alpha }_1,\dots ,{\alpha }_u$ and ${\beta }_1,\dots ,{\beta }_v$, a mixed sum $\mathrm{\Phi }={\alpha }_1{P}_4(x_1)+\cdots +{\alpha }_u{P}_4(x_u)+{\beta }_1{P}_8(y_1)+\cdots +{\beta }_v{P}_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $\mathrm{\Phi }=N$ has an integer solution for any nonnegative integer $N$. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the “$61$-theorem” is proved, which states that an arbitrary mixed sum of generalized $4$- and $8$-gonal numbers is universal if and only if it represents the integers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $15$, $18$, $20$, $30$, $60$, and $61$.