Rafael Tesoro
Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erdős and Harzheim about these sets.
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Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=μM+1$ with $μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,η,δ$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $η\equiv1\left(mod\,2\right)$ and $δ\equiv0,1$ or $5\left(mod\,6\right)$, and if $δ\equiv0\left(m…
We prove that the rational elliptic curve y^2 = x^3 - n^2x satisfies the full Birch and Swinnerton-Dyer conjecture for at least 41.9% of positive squarefree integers n equal to 1, 2, or 3 mod 8, and that it satisfies the regular BSD conjecture for at least 55.9% of positive squarefree integers n equal to 5, 6, or 7 mod 8. In particular, at least 55.9% of positive squarefree integers equal to 5, 6, or 7 mod 8 are congruent numbers. These proofs complete an argument started by Tian, Yuan, and Zhang.
We prove that a positive proportion of squarefree integers are congruent numbers such that the canonical height of the lowest non-torsion rational point on the corresponding elliptic curve satisfies a strong lower bound.
In this paper, based on an ideal of Tian, we establish a new sufficient condition for a positive integer to be a congruent number in terms of Legendre symbols of prime factors of the positive integer. Our criterion generalizes previous criterions of Heegner, and Birch--Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving our criterion is to give formulae for the analytic Tate--Shafarevich number in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the genera…
$2$-Selmer groups, $2$-class groups, and congruent numbers
本文研究了特定平方自由整数为同余数的必要条件，通过相关虚二次域类数的可除性性质，探讨了2-类组与同余数之间的关系，并给出了非同余数的数量下界。
Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < θ< π$ has rational cosine, say $\cos (θ)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive integer $n$ is called a $(\mathbb K,θ)$-congruent number if there is a triangle, called the $(\mathbb K,θ, n)$-triangles, with sides in $\mathbb K$ having $θ$ as an angle and $nα_θ$ as area, where ${α_θ}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,θ)$-congruent number elliptic curve $E_{n,θ}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb K$. Denote the squarefree part of posi…
A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.
A necessary condition for a congruent number of the form $8k+3$
研究8k+3形式同余数的必要条件，通过修改的Rédei矩阵探讨二次体类数2部分的模2次方关系。
In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive integers $a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ax^2+by^2+cT_z$ represents sufficiently large integers and establish similar results for the forms $ax^2…
To each i, j belonging to some set of integers, attach the integer a(i,j). Are there integers x(i) such that x(j)-x(i) is congruent to a(i,j) mod (i,j)? A necessary condition is that a(i,j)+a(j,k) be congruent to a(i,k) mod (i,j,k). This condition is sufficient.
We study integral points on the quadratic twists $\mathcal{E}_D:y^2=x^3-D^2x$ of the congruent number curve. We give upper bounds on the number of integral points in each coset of $2\mathcal{E}_D(\mathbb{Q})$ in $\mathcal{E}_D(\mathbb{Q})$ and show that their total is $\ll (3.8)^{\mathrm{rank} \mathcal{E}_D(\mathbb{Q})}$. We further show that the average number of non-torsion integral points in this family is bounded above by $2$. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations $aX^2-bY^2=d$, $bY^2-cZ^2=d$ for pairwise coprime positive inte…
In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p congruent to 7 modulo 8. In addition to the above, assume that one of the following holds: Either (i) n^2-p>0 and the positive integer is a prime, Or (ii) n^2-p<0 and the positive integer N=-m=-(n^2-p) is a prime. Then the diophantine equation x^4 +2nx^2y^2+my^4=z^2 has no positive integer solutions. The method of proof is elementary in that it only uses co…
A variant of the congruent number problem
本文针对固定角余弦为$\pm \sqrt{2}/2$的情形，研究$θ$-同余数问题的一个变体，该问题与特定椭圆曲线的有理点存在性相关。
A positive integer that is the area of some rational right triangle is called a congruent number. In an algebraic point of view, being a congruent number means satisfying a system of equations. As early as the 1800s, it is understood that if $n$ is a congruent number, then the equation $nm^2 = uv(u^2 - v^2)$ has a solution in $\mathbb{Z}$. Using the relation between congruent numbers and elliptic curves $E_n: y^2 = x^3 - n^2 x$ which was established in the 1900s, we will prove that the converse of this two century-old result holds as well. In addition to this, we present another proof of the c…
A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its product is congruent to $μ(p-1)$ modulo $p$, where $μ$ is the Möbius function. In this paper, we will generalize these two interesting congruences and give the congruences of the sum and the product of integers with the same index modulo $n$.
A positive integer $N$ is called a $θ$-congruent number if there is a $\ta$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $θ$ and its area is $N \sqrt{r^2-s^2}$, where $θ\in (0, π)$, $\cos(θ)=s/r$, and $0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\ta$-congruent number if and only if the elliptic curve $E_N^\ta: y^2=x (x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\neq 1,2,3,6$ is a $\ta$-congruent number if and only if rank of …
Characterization of order structures avoiding three-term arithmetic progressions
本文完全刻画了非负整数、整数及有理数集合上避免单调三项等差数列的混沌全序的序结构。
A partition $α$ is said to contain another partition (or pattern) $μ$ if the Ferrers board for $μ$ is attainable from $α$ under removal of rows and columns. We say $α$ avoids $μ$ if it does not contain $μ$. In this paper we count the number of partitions of $n$ avoiding a fixed pattern $μ$, in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $μ$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic $p$-groups. We f…
Hasse principle violation for algebraic families of del Pezzo surfaces of degree 4 and hyperelliptic curves of genus congruent to 1 modulo 4
本文构造了无限多个显式单参数代数族，包含次数为4的del Pezzo曲面和亏格模4余1的超椭圆曲线，证明其每个K成员都违反Hasse原则，并由Brauer-Manin障碍解释。
We describe the structure of a set of integers $A$ of positive density $δ$, such that $A+A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the values $δ_0=1/4-\frac{2}{π^2}=0.0473...$ and $δ_1=1/18$.
Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
本文在秩为零的同余数曲线上分类了最简三次域中给定横坐标的点，通过群迹排除和四次方程得到有限参数对，并构造了无穷多个域上的点。
Let $p_t(a,b;n)$ denote the number of partitions of $n$ such that the number of $t$ hooks is congruent to $a \bmod{b}$. For $t\in \{2, 3\}$, arithmetic progressions $r_1 \bmod{m_1}$ and $r_2 \bmod{m_2}$ on which $p_t(r_1,m_1; m_2 n + r_2)$ vanishes were established in recent work by Bringmann, Craig, Males, and Ono using the theory of modular forms. Here we offer a direct combinatorial proof of this result using abaci and the theory of $t$-cores and $t$-quotients.
Any positive integer $n$ other than 10 with abundancy index 9/5 must be a square with at least 6 distinct prime factors, the smallest being 5. Further, at least one of the prime factors must be congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime power factorization of $n$.
We reformulate the $3x+1$ conjecture by restricting attention to numbers congruent to $2$ (mod $3$). This leads to an equivalent conjecture for positive integers that reveals new aspects of the dynamics of the $3x+1$ problem. Advantages include a governing function with particularly simple mapping properties in terms of partitions of the set of integers. We use the refined conjecture to obtain a new characterization of $3x+1$ trajectories that shows a special role played by numbers congruent to $2$ or $8$ (mod $9$). We construct an accelerated iteration whose long-term behavior involves only t…
Pach and Palincza proved the following generalization of Ellenberg and Gijswijt's bound for the size of $k$-term arithmetic progression-free subsets, where $k\in \{4,5,6\}$: Let $m>0$ be an integer such that $6$ divides $m$ and let $k\in \{4,5,6\}$. Then $$ r_k({\mathbb Z}_{m}^n)\leq (0.948m)^n $$ if $n$ is sufficiently large. Building on the proof technique of Pach and Palincza's upper bound we generalize the Ellenberg and Gijswijt's bound in the following way: Let $p>2$ be any integer and let $q>2$ be a prime. Suppose that $p\leq q$. Then the there exists an $n_0\in \mathbb N$ integer and a …
Let K be an algebraic number field and O_K be its ring of integers. Let S_K be the set of elements in O_K which are sums of squares in O_K and s(O_K) the minimal number of squares necessary to represent -1in O_K. Let g( S_K ) be the smallest positive integer t such that every element in S_K is a sum of t squares in O_K. Here K is generated over field of rational number by square root of m and -n , where m congruent 3 mod 4 and n congruent 1 mod 4 are two distinct positive square free integers, we prove that $ S_K= O_K. We also prove that g(O_ K) less or equals to s(O_K)+1 or s(O_K)+2. Applying…
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(ł_1, ł_2, \dots, ł_k) \in \Z_{> 0}^k$ such that $n = ł_1 + ł_2 + \dots + ł_k$. Our goal is to enumerate those compositions whose parts $ł_1, ł_2, \dots, ł_k$ avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer $k$, the resulting compositions are counted by a seq…
A large integer is a sum of two prime avoiding numbers
本文证明了任意大整数可表示为两个正整数之和，且这两个数与最近素数的距离至少为 $\ln N (\ln\ln N)^δ$。
Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions
本文将不含三项等比数列的最大密度问题推广到非交换的 Hurwitz 四元数序，给出贪心集合密度的精确公式，并推导最优上密度的上下界。
Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions
本研究给出数字受限整数集不含k项算术级数的条件，通过大规模搜索找到不含4项与10项算术级数且调和和优于传统贪心构造的整数集，可高效计算相关集合与调和和。
For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log N)(\log\log N)^{1/325565}, $$ for $i=1,2$. This improves the corresponding "trivial" statement where only $F(n_i)\gg \log N$ is required.
Integer Partitions With Restricted Distinct Parts
本文研究正整数分拆为互不相同部分且禁止特定模 $t$ 同余类的计数函数 $Q_{t}^{s}(n)$，并利用 $q$-级数与 theta 函数恒等式证明了若干 Ramanujan 型同余式。
Integral Quadratic Forms Avoiding Arithmetic Progressions
该研究证明对任意正整数k，存在正定对角四元整二次型恰好表示除k个算术级数中数之外的所有正整数，并确定了k=1时所有满足该性质的二次型，丰富了数论中整二次型表示理论。
Avoiding algebraic integers of bounded house in orbits of rational functions over cyclotomic closures
本文证明了在数域的分圆闭包上，许多有理函数的取值或轨道中house有界的代数整数集合是有限的，推广了Ostafe关于避免单位根的结果。
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of partitions avoiding an arbitrary set of "patterns".
Two well studied Ramsey-theoretic problems consider subsets of the natural numbers which either contain no three elements in arithmetic progression, or in geometric progression. We study generalizations of this problem, by varying the kinds of progressions to be avoided and the metrics used to evaluate the density of the resulting subsets. One can view a 3-term arithmetic progression as a sequence $x, f_n(x), f_n(f_n(x))$, where $f_n(x) = x + n$, $n$ a nonzero integer. Thus avoiding three-term arithmetic progressions is equivalent to containing no three elements of the form $x, f_n(x), f_n(f_n…
Integral zeros of quadratic polynomials avoiding sublattices
本文研究避开有限指数子格的整系数二次多项式的非平凡零点，证明其存在范数有界零点并给出显式上界，同时将结果应用于整数格中向量夹角分布问题。
Let $p$ be a prime congruent to 1 or 3 modulo 8 so that the equation $p=a^2+2b^2$ is solvable in integers. In this paper, we obtain closed-form expressions for $a$ and $b$ in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.
Let $n\ge 2$ be an integer and let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given $\mathbb F_q$-affine hyperplanes $\mathcal A_1, \ldots, \mathcal A_n$ of $\mathbb F_{q^n}$ in general position, we study the existence and distribution of primitive elements of $\mathbb F_{q^n}$, avoiding each $\mathcal A_i$. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field Q(λ) (depending on n) and an explicit point P_λof infinite order in the Mordell-Weil group of the elliptic curve Y^2=X^3-n^2*X over Q(λ).
Let $N>1$ be an integer, and let $Γ= Γ_0 (N) \subset \SL_4 (\Z)$ be the subgroup of matrices with bottom row congruent to $(0,0,0,*)\mod N$. We compute $H^5 (Γ; \C) $ for a range of $N$, and compute the action of some Hecke operators on many of these groups. We relate the classes we find to classes coming from the boundary of the Borel-Serre compactification, to Eisenstein series, and to classical holomorphic modular forms of weights 2 and 4.
Conflict-avoiding codes and near-primitive roots
本文研究冲突避免码中涉及$4$模$d$乘法阶的量的分布，利用Artin本原根猜想技术，得到奇素数和奇数情形下的结果，并给出该量与其他数学对象的联系。
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