Finding Almost Squares

Three consecutive almost squares

Given a positive integer $n$, we let ${\rm sfp}(n)$ denote the squarefree part of $n$. We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} \leq 150$ by relating the problem to finding integral points on elliptic curves. We also prove that there are infinitely many $n$ for which \[ \max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm sfp}(n+2) \} < n^{1/3}. \]

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.

Lucas sequences whose nth term is a square or an almost square

(Below, \Box means "perfect square") Let $P$ and $Q$ be non-zero integers. The Lucas sequence $\{U_n(P,Q)\}$ is defined by $U_0=0$, $U_1=1$, $U_n=P U_{n-1}-Q U_{n-2}$, $(n \geq 2)$. Historically, there has been much interest in when the terms of such sequences are perfect squares (or higher powers). Here, we summarize results on this problem, and investigate for fixed $k$ solutions of $U_n(P,Q)= k\Box$, $(P,Q)=1$. We show finiteness of the number of solutions, and under certain hypotheses on $n$, describe explicit methods for finding solutions. These involve solving finitely many Thue-Mahler e…

几乎素数与可表示为两平方和加1的素数

Almost primes and primes that are sums of two squares plus 1

该研究用向量筛框架（含半线性筛、线性筛）得出满足p-1为两平方和且p+2素因子个数有界的素数p≤x的下界。

On Linnik's conjecture: sums of squares and microsquares

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than (log n)^{1+e} (any e>0). This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick and Sarnak concerning nearest neighbour distances between normalised integral points on the sphere.

Sums of almost equal squares of primes

We study the representations of large integers $n$ as sums $p_1^2 + ... + p_s^2$, where $p_1,..., p_s$ are primes with $| p_i - (n/s)^{1/2} | \le n^{θ/2}$, for some fixed $θ< 1$. When $s = 5$ we use a sieve method to show that all sufficiently large integers $n \equiv 5 \pmod {24}$ can be represented in the above form for $θ> 8/9$. This improves on earlier work by Liu, Lü and Zhan, who established a similar result for $θ> 9/10$. We also obtain estimates for the number of integers $n$ satisfying the necessary local conditions but lacking representations of the above form with $s = 3, 4$. When $…

Farmer Ted Goes Natural

A traditional "Farmer Ted" calculus problem is to minimize the perimeter of a rectangular chicken coop given the area N, so that as little as possible will be spent on the fencing. But what if N is an integer, and we are only allowed to consider rectangles with integer side lengths? Often it will be more cost-effective to build a coop with area smaller than N, where the measure of cost-effectiveness is the ratio of the area to the perimeter. Those numbers N that are the areas of rectangles that are more cost-effective than any smaller rectangle are dubbed "almost-squares", in deference to our …

所有平方数之间的殆素数

Almost primes between all squares

本文证明了对于所有 $n\geq 1$，在 $n^2$ 和 $(n+1)^2$ 之间存在至多有 4 个素因子的数，这是首个对所有 $n$ 成立的此类结果，方法依赖 Sorenson 和 Webster 的计算、显式线性筛法及新证明的显式 Kuhn 加权筛法。

在平方乘积中寻找平方

Finding Squares in a Product of Squares

本文研究丢番图方程 ∏(k²+1)=b² 的正整数解，并运用解析数论方法进行求解。

Algorithms for Finding Almost Irreducible and Almost Primitive Trinomials

Consider polynomials over ${\rm GF}(2)$. We describe efficient algorithms for finding trinomials with large irreducible (and possibly primitive) factors, and give examples of trinomials having a primitive factor of degree $r$ for all Mersenne exponents $r = \pm 3 \bmod 8$ in the range $5 < r < 10^7$, although there is no irreducible trinomial of degree $r$. We also give trinomials with a primitive factor of degree $r = 2^k$ for $3 \le k \le 12$. These trinomials enable efficient representations of the finite field ${\rm GF}(2^r)$. We show how trinomials with large primitive factors can be used…

On squares of sums of three cubes

We show that almost every positive integer can be expressed as a sum of four squares of integers represented as the sums of three positive cubes.

Smooth squarefree and square-full integers in arithmetic progressions

We obtain new lower bounds on the number of smooth squarefree integers up to $x$ in residue classes modulo a prime $p$, relatively large compared to $x$, which in some ranges of $p$ and $x$ improve that of A. Balog and C. Pomerance (1992). We also estimate the smallest squarefull number in almost all residue classes modulo a prime $p$.

On the Waring-Goldbach Problem for One Square and Five Cubes

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^3 \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_6$ and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with $\mathcal{P}_{36}$ in place of $\mathcal{P}_6$.

Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12\leqslant b\leqslant 35$ and for every sufficiently large odd integer $N$, the equation \begin{equation*} N=x^2+p_1^3+p_2^3+p_3^3+p_4^3+p_5^4+p_6^b \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(b)}$ and the other variables primes, where $r(b)$ is defined in the Theorem. This result constitutes an improvement upon that of Lü and Mu.

Coordinate Distribution of Gaussian Primes

We study the problem of writing Gaussian primes as the sum of two squares, both of which are interesting arithmetically, in particular, when one is the square of a prime and the other the square of an almost-prime.

On Waring-Goldbach Problem for Squares, Cubes and Higher Powers

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan for ternary admissible exponent. Moreover, we use the refined admissible exponent to prove that, for $3\leqslant k\leqslant 14$ and for every sufficiently large even integer $n$, the following equation \begin{equation*} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{equation*} is solvable with $x$ being an almost-prime $\mathcal{P}_{r(k)}$ and the other variables primes, where $r(k)$ is defined in Theorem. This result constitutes a deepening…

Sums of primes and squares of primes in short intervals

Let $\mathbf H_2$ denote the set of even integers $n \not\equiv 1 \pmod 3$. We prove that when $H \ge X^{0.33}$, almost all integers $n \in \mathbf H_2$, $X < n \le X + H$ can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.

A basis of the group of primitive almost pythagorean triples

Let $m$ be a fixed square-free positive integer, then equivalence classes of solutions of Diophantine equation $x^2+m\cdot y^2=z^2$ form an infinitely generated abelian group under the operation induced by the complex multiplication. A basis of this group is constructed here using prime ideals and the ideal class group of the field $\mathbb Q (\sqrt{-m})$.

Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials

Square roots $s$ of sums of $M$ consecutive integer squares starting from $a^{2}\geq1$ are integers if $M\equiv0,9,24$ or $33(mod\,72)$; or $M\equiv1,2$ or $16(mod\,24)$; or $M\equiv11(mod\,12)$ and cannot be integers if $M\equiv3,5,6,7,8$ or $10(mod\,12)$. Finding all solutions with $s$ integer requires to solve a Diophantine quadratic equation in variables $a$ and $s$ with $M$ as a parameter. If $M$ is not a square integer, the Diophantine quadratic equation in variables $a$ and $s$ is transformed into a generalized Pell equation whose form depends on the $M(mod\,4)$ congruent value, and who…

Linnik微平方问题中几乎所有整数的最优尺度

The optimal scale in Linnik's microsquare problem for almost all integers

本文证明Linnik微平方问题中几乎所有整数的最优尺度为$\sqrt{\log n}$，改进Wooley界，并给出奇数表示为无平方因子数与二的幂之和的充要条件。

On the Generation of Pythagorean Triples and Representation of Integers as a Difference of Two Squares

The general formulas for finding the quantity of all primitive and nonprimitive triples generated by the given number x have been proposed. Also the formulas for finding the complete quantity of the representations of the integers as a difference of two squares have been obtained.

Finding Sums of Four Squares via Complex Continued Fractions

The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain …

On the symmetry of the Liouville function in almost all short intervals

We prove a kind of "almost all symmetry" result for the Liouville function $λ(n):=(-1)^{Ω(n)}$, giving non-trivial bounds for its "symmetry integral", say $I_λ(N,h)$ : we get $I_λ(N,h)\ll NhL^3+Nh^{21/20}$, with $L:=\log N$. We also give similar results for other related arithmetic functions, like the Möbius function $μ(n)$ ($=λ(n)$ on square-free $n$).

Ternary Sums of Squares and Triangular Numbers

For any integer $x$, let $T_x$ denote the triangular number $\frac{x(x+1)}{2}$. In this paper we give a complete characterization of all the triples of positive integers $(α, β, γ)$ for which the ternary sums $αx^2 +βT_y + γT_z$ represent all but finitely many positive integers. This resolves a conjecture of Kane and Sun \cite[Conjecture 1.19(i)]{KS08} and complete the characterization of all almost universal ternary mixed sums of squares and triangular numbers.

Squarefree values of polynomials over the rational function field

We study representation of square-free polynomials in the polynomial ring F[t] over a finite field F by polynomials in F[t][x]. This is a function field version of the well-studied problem of representing squarefree integers by integer polynomials, where it is conjectured that a separable polynomial f(x) with integer coefficients takes infinitely many squarefree values, barring some simple exceptional cases, in fact that the integers n for which f(n) is squarefree have a positive density. We show that if f(x) in F[t][x] is separable, with square-free content, of bounded degree and height, then…

关于近平坦SIC-POVMs的一个猜想

A Conjecture on Almost Flat SIC-POVMs

本文探讨具有反单位元对称性的SIC-POVMs的恒等式是否能确定Stark单位，发现其不足以完全确定但可能影响较小。

短区间内平方满数数量的方差

Variance of Squarefull Numbers in Short Intervals

本文通过研究短区间内平方满数数量的方差，证明了对任意 $0 < θ < 1/2$，几乎所有短区间 $(x, x + x^{1/2 + θ}]$ 中平方满数的数量约为 $\ rac{ζ(3/2)}{2 ζ(3)} x^θ$。

On almost universal mixed sums of squares and triangular numbers

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…

关于在几乎所有素数模下有根的多项式

On polynomials with roots modulo almost all primes

该研究定义了在几乎所有素数模下有根但无整数根的首一整系数多项式为例外多项式，分类了与不可约首一二次多项式乘积为例外多项式的不可约首一整系数多项式，并构造了指定因式形式的例外多项式。

$\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$ 非均匀商中轨形轨道几乎平方的稠密性

Density of almost squares of horospherical orbits in non-uniform quotients of $\operatorname{SL}_2(\mathbb{R})\times\operatorname{SL}_2(\mathbb{R})$

本文证明在SL2(R)×SL2(R)的非均匀格商空间中，具有稠密轨道的标准轨形子群在几乎平方点处的轨道仍稠密，通过有效追踪周期轨道并利用Weyl不等式及齐次动力学工具实现。

Sets whose differences avoid squares modulo m

We prove that if $\varepsilon(m)\to 0$ arbitrarily slowly, then for almost all $m$ and any $A\subset\mathbb{Z}_m$ such that $A-A$ does not contain non-zero quadratic residues we have $|A|\leq m^{1/2-\varepsilon(m)}.$

A probabilistic technique for finding almost-periods of convolutions

We introduce a new probabilistic technique for finding 'almost-periods' of convolutions of subsets of groups. This gives results similar to the Bogolyubov-type estimates established by Fourier analysis on abelian groups but without the need for a nice Fourier transform to exist. We also present applications, some of which are new even in the abelian setting. These include a probabilistic proof of Roth's theorem on three-term arithmetic progressions and a proof of a variant of the Bourgain-Green theorem on the existence of long arithmetic progressions in sumsets A+B that works with sparser subs…

Square Eulerian Quadruples

We consider the problem of finding 4 rational squares, such that the product of any two plus the sum of the same two always gives a square. We give some historical background and exhibit one such quadruple.

The Group of Primitive Almost Pythagorean Triples

We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and give a complete analysis when $q\in\{2,3,5,6\}$.