arXiv subjects
Publications and source records attributed to Byeong-Kweon Oh.
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.
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 $α_1,\dots,α_u$ and $β_1,\dots,β_v$, a mixed sum $Φ=α_1P_4(x_1)+\cdots+α_uP_4(x_u)+β_1P_8(y_1)+\cdots+β_vP_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $Φ=N$ 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 $61$.
Let $p$ be a prime. We define $S(p)$ the smallest number $k$ such that every positive integer is a sum of at most $k$ squares of integers that are not divisible by $p$. In this article, we prove that $S(2)=10$, $S(3)=6$, $S(5)=5$, and $S(p)=4$ for any prime $p$ greater than $5$. In particular, it is proved that every positive integer is a sum of at most four squares not divisible by $5$, except the unique positive integer $79$.
For positive integers $a,b,c$, and an integer $n$, the number of integer solutions $(x,y,z) \in \mathbb Z^3$ of $a \frac{x(x-1)}{2} + b \frac{y(y-1)}{2} + c \frac{z(z-1)}{2} = n$ is denoted by $t(a,b,c;n)$. In this article, we prove some relations between $t(a,b,c;n)$ and the numbers of representations of integers by some ternary quadratic forms. In particular, we prove various conjectures given by Z. H. Sun in \cite{s}.
Let $f(x,y)=ax^2+bxy+cy^2$ be a binary quadratic form with integer coefficients. For a prime $p$ not dividing the discriminant of $f$, we say $f$ is completely $p$-primitive if for any non-zero integer $N$, the diophantine equation $f(x,y)=N$ has always an integer solution $(x,y)=(m,n)$ with $(m,n,p)=1$ whenever it has an integer solution. In this article, we study various properties of completely $p$-primitive binary quadratic forms. In particular, we give a necessary and sufficient condition for a definite binary quadratic form $f$ to be completely $p$-primitive.
An integer of the form $P_8(x)=3x^2-2x$ for some integer $x$ is called a generalized octagonal number. A quaternary sum $Φ_{a,b,c,d}(x,y,z,t)=aP_8(x)+bP_8(y)+cP_8(z)+dP_8(t)$ of generalized octagonal numbers is called {\it universal} if $Φ_{a,b,c,d}(x,y,z,t)=n$ has an integer solution $x,y,z,t$ for any positive integer $n$. In this article, we show that if $a=1$ and $(b,c,d)=(1,3,3), (1,3,6), (2,3,6), (2,3,7)$ or $(2,3,9)$, then $Φ_{a,b,c,d}(x,y,z,t)$ is universal. These were conjectured by Sun in \cite {sun}. We also give an effective criterion on the universality of an arbitrary sum $a_1 P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ of generalized octagonal numbers, which is a generalization of "$15$-theorem" of Conway and Schneeberger.
A quadratic polynomial $Φ_{a,b,c}(x,y,z)=x(ax+1)+y(by+1)+z(cz+1)$ is called universal if the diophantine equation $Φ_{a,b,c}(x,y,z)=n$ has an integer solution $x,y,z$ for any non negative integer $n$. In this article, we show that if $(a,b,c)=(2,2,6), (2,3,5)$ or $(2,3,7)$, then $Φ_{a,b,c}( x,y,z)$ is universal. These were conjectured by Sun in \cite {Sun}.
For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski-Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be written as a constant multiple of a product of local densities which are easily computable. In this article, we consider the case when the spinor genus of $f$ contains only one class. In this case the above also holds if $k$ is not contained in a set of finite number of square classes which are easily computable (see, for example, \cite{sp1} and \cite {sp2}). By using this fact, we prove some extension of the results given in both \cite {cl} on the representations of generalized Bell ternary forms and \cite {be} on the representations of ternary quadratic forms with some congruence conditions.
For two positive definite integral ternary quadratic forms $f$ and $g$ and a positive integer $n$, if $n\cdot g$ is represented by $f$ and $n\cdot dg=df$, then the pair $(f,g)$ is called a representable pair by scaling $n$. The set of all representable pairs in $\text{gen}(f)\times \text{gen}(g)$ is called a genus-correspondence. Jagy conjectured that if $n$ is square free and the number of spinor genera in the genus of $f$ equals to the number of spinor genera in the genus of $g$, then such a genus-correspondence respects spinor genus in the sense that for any representable pairs $(f,g), (f',g')$ by scaling $n$, $f' \in \text{spn}(f)$ if and only if $g' \in \text{spn}(g)$. In this article, we show that by giving a counter example, Jagy's conjecture does not hold. Furthermore, we provide a necessary and sufficient condition for a genus-correspondence to respect spinor genus.
Let $f$ be a positive definite ternary quadratic form. We assume that $f$ is non-classic integral, that is, the norm ideal of $f$ is $\z$. We say $f$ is {\it strongly $s$-regular } if the number of representations of squares of integers by $f$ satisfies the condition in Cooper and Lam's conjecture in \cite {cl}. In this article, we prove that there are only finitely many strongly $s$-regular ternary forms up to equivalence if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly $207$ non-classic integral strongly $s$-regular ternary forms that represent one (see Tables 1 and 2). This result might be considered as a complete answer to a natural extension of Cooper and Lam's conjecture.
Let $L$ be a positive definite (non-classic) ternary $\z$-lattice and let $p$ be a prime such that a $\frac 12\z_p$-modular component of $L_p$ is nonzero isotropic and $4\cdot dL$ is not divisible by $p$. For a nonnegative integer $m$, let $\mathcal G_{L,p}(m)$ be the genus with discriminant $p^m\cdot dL$ on the quadratic space $L^{p^m}\otimes \q$ such that for each lattice $T \in \mathcal G_{L,p}(m)$, a $\frac 12\z_p$-modular component of $T_p$ is nonzero isotropic, and $T_q$ is isometric to $(L^{p^m})_q$ for any prime $q$ different from $p$. Let $r(n,M)$ be the number of representations of an integer $n$ by a $\z$-lattice $M$. In this article, we show that if $m \le 2$ and $n$ is divisible by $p$ only when $m=2$, then for any $T \in \mathcal G_{L,p}(m)$, $r(n,T)$ can be written as a linear summation of $r(pn,S_i)$ and $r(p^3n,S_i)$ for $S_i \in \mathcal G_{L,p}(m+1)$ with an extra term in some special case. We provide a simple criterion on when the extra term is necessary, and we compute the extra term explicitly. We also give a recursive relation to compute $r(n,T)$, for any $T \in \mathcal G_{L,p}(m)$, by using the number of representations of some integers by lattices in $\mathcal G_{L,p}(m+1)$ for an arbitrary integer $m$.
Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$, denoted by $\text{gen}(f)$, is indistinguishable by squares if for any integer $n$, $r(n^2,f)=r(n^2,f')$ for any quadratic form $f' \in \text{gen}(f)$. We find some non trivial genera of ternary quadratic forms which are indistinguishable by squares. We also give some relation between indistinguishable genera by squares and the conjecture given by Cooper and Lam, and we resolve their conjecture completely.
G.L. Watson \cite{watson1, watson2} introduced a set of transformations, called Watson transformations by most recent authors, in his study of the arithmetic of integral quadratic forms. These transformations change an integral quadratic form to another integral quadratic form with a smaller discriminants, but preserve many arithmetic properties at the same time. In this paper, we study the change of class numbers of positive definite ternary integral quadratic formula along a sequence of Watson transformations, thus providing a new and effective way to compute the class number of positive definite ternary integral quadratic forms. Explicit class number formulae for many genera of positive definite ternary integral quadratic forms are derived as illustrations of our method.
In this paper, we study the representations of integral quadratic polynomials. Particularly, it is shown that there are only finitely many equivalence classes of positive ternary universal integral quadratic polynomials, and that there are only finitely many regular ternary triangular forms. A more general discussion of integral quadratic polynomials over a Dedekind domain inside a global field is also given.
For any $m\ge3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ be positive integers. For every non negative integer $n$, if there are integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$, then the quadruple $(k;a,b,c)$ is said to be {\it universal}. Sun gave in \cite{s1} all possible quadruple candidates that are universal and proved some quadruples to be universal (see also \cite{gs}). He remains the following quadruples $(5,1,1,k)$ for $k=6,8,9,10$, $(5,1,2,8)$, and $(5,1,3,s)$ for $7 \le s \le 8$ as candidates and conjectured the universality of them. In this article we prove that the remaining 7 quadruples given above are, in fact, universal.
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if T_m=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p^2=x^2+8(y^2+z^2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2T_m (m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.