Almost universal mixed sums of squares and polygonal numbers

For each integer $m\ge3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. Given positive integers $a,b,c,k$ and an odd prime number $p$ with $p\nmid c$, we employ the theory of ternary quadratic forms to determine completely when the mixed sum $ax^2+by^2+cP_{p^k+2}(z)$ represents all but finitely many positive integers.Comment: 20 page

DAC sets of Researcher B.

<p>DAC sets of Researcher B.</p

Relationships among the DAC sets of Researcher B.

<p>Relationships among the DAC sets of Researcher B.</p

<i>W</i>_<i>ij</i> matrix.

<p><i>W</i>_<i>ij</i> matrix.</p

Average weighted-mean DIRD values of 3 kinds of universities.

<p>Average weighted-mean DIRD values of 3 kinds of universities.</p

Mathematics professors' top 10 favorite research areas.

<p>Mathematics professors' top 10 favorite research areas.</p

Classic DIRD and weighted-mean DIRD.

<p>Classic DIRD and weighted-mean DIRD.</p

Transfer the DACs to DAC sets.

<p>Transfer the DACs to DAC sets.</p

Composition of the data set.

<p>Composition of the data set.</p

DIRD comparisons among “985”, “211” and “normal” universities.

<p>DIRD comparisons among “985”, “211” and “normal” universities.</p