Byeong Moon Kim, Dayoon Park
In this paper, we show that for a fixed rank $n$, there are only finitely many $m$ for which there is a regular $m$-gonal form of rank $n$ and determine every type of the (generalized) regular $m$-gonal form for every sufficiently large $m$.
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The maximal $n$ which admits a non-almost regular $m$-gonal form of rank $n$ is $4$. In this article, we consider the $m$-gonal forms of rank $4$ which is not almost regular.
It is known that any $m$-gonal form of $\rank n \ge 5$ is almost regular. In this article, we study the sufficiently large integers which are represented by (almost regular) $m$-gonal forms of $\rank n \ge 6$.
It is known that any $m$-gonal form of rank $n \ge 5$ is almost regular. On the other words, any $m$-gonal form of rank $n \ge 5$ represents every sufficiently large integer which is locally represented. In this article, we study the sufficiently large integers which are represented by (almost regular) $m$-gonal forms of rank $n \ge 5$.
In this article, we consider the representation of $m$-gonal forms over $\mathbb N_0$. We show that any $m$-gonal forms over $\mathbb N_0$ of rank $\ge 5$ is almost regular and ponder the sufficiently large integers which are indeed represented over $\mathbb N_0$ among the integers which are locally represented. And as its consequential result, we prove a finiteness theorem for universal (original polygonal number version's) $m$-gonal forms over $\mathbb N_0$.
In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.
In this paper, we study the set of positive integers that characterize the universality of $m$-gonal form.
In this paper, we show that for any $m$-gonal form $F_m(\mathbf x)$ with $m \ge 12$ which represents every positive integer up to $m-4$, by putting together only unary $m$-gonal form, we may complete an universal form.
A Formula For All K-Gonal Numbers that Are Centered
固定一个正整数k。本文给出了所有中心k-多边形的k-多边形数的一个封闭公式。
In this paper, we classify the $(a_1,a_2,a_3,a_4,a_5)$ for which the universality of an $m$-gonal form $F_m(\mathbf x)$ having its first five coefficients as $(a_1,a_2,a_3,a_4,a_5)$ is characterized as the representability of positive integers up to $m-4$ and see its applications.
Regular ternary sums of generalized polygonal numbers
研究正则三元和的广义多边形数，确定存在一个常数C，当m大于C时，不存在这样的和。
Inspired by Dickson's classification of regular ternary quadratic forms, we prove that there are no primitive regular $m$-gonal forms when $m$ is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next prime by the product of the previous primes, a question of independent interest.
Tight universality of $m$-gonal forms with minimal criterion sets
本文确定了紧致 $\mathcal{T}(n)$-普适 $m$ 角型的最小判定集，给出了 $n\geq8$ 时除两个例外情形的完整刻画，并证明 $m$ 的上界最优。
For a natural number $m$, generalized $m$-gonal numbers are those numbers of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in \mathbb Z$. In this paper we establish conditions on $m$ for which the ternary sum $p_m(x)+p_m(y)+p_m(z)$ is almost universal.
A positive-definite integral quadratic form is called regular if it represents every positive integer which is locally represented. In this article, we classify all regular diagonal quadratic forms of rank greater than 3.
A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic forms and prove the odd-regularities of all but 6 candidates.
The goal of this note is to provide an analysis of the positive integers that are represented everywhere locally, but not globally, by each of the 29 spinor regular positive definite integral ternary quadratic forms that are not regular.
Extending the notion of regularity introduced by Dickson in 1939, a positive definite ternary integral quadratic form is said to be spinor regular if it represents all the positive integers represented by its spinor genus (that is, all positive integers represented by any form in its spinor genus). Jagy conducted an extensive computer search for primitive ternary quadratic forms that are spinor regular, but not regular, resulting in a list of 29 such forms. In this paper, we will prove that there are no additional forms with this property.
An integral quadratic form is called strictly $n$-regular if it primitively represents all quadratic forms in $n$ variables that are primitively represented by its genus. For any $n \geq 2$, it will be shown that there are only finitely many similarity classes of positive definite strictly $n$-regular integral quadratic forms in $n + 4$ variables. This extends the recent finiteness results for strictly regular quaternary quadratic forms by Earnest-Kim-Meyer (2014).
A (positive definite and integral) quadratic form $f$ is called regular if it represents all integers that are locally represented. It is known that there are only finitely many regular ternary quadratic forms up to isometry. However, there are infinitely many equivalence classes of regular quadratic forms of rank $n$ for any integer $n$ greater than or equal to $4$. A regular quadratic form $f$ is called new if there does not exist a proper subform $g$ of $f$ such that the set of integers that are represented by $g$ is equal to the set of integers that are represented by $f$. In this article,…
An integer of the form $T_x=\frac{x(x+1)}2$ for some positive integer $x$ is called a triangular number. A ternary triangular form $aT_{x}+bT_{y}+cT_{z}$ for positive integers $a,b$ and $c$ is called regular if it represents every positive integer that is locally represented. In this article, we prove that there are exactly 49 primitive regular ternary triangular forms.
A triangular form is defined to be an integer-valued quadratic polynomial of the form $a_1P_3(x_1)+a_2P_3(x_2)+\cdots+a_kP_3(x_k)$ where $a_i's$ are positive integers and $P_3(x)=x(x+1)/2$. A triangular form is called regular if it represents all positive integers which are locally represented. In this article, we find all regular triangular forms of more than three variables.
A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any integer $k \ge 2$, there are only finitely many isometry classes of strongly $s$-regular quadratic forms with rank $k$ if the minimum of the nonzero squares that are represented by them is fixed.
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