A positive integer $n$ is called a practical number if every positive integer less than or equal to $n$ can be expressed as a sum of distinct positive divisors of $n$. In this paper, we study an open conjecture proposed by Wang and Sun concerning the quadratic representations of practical numbers. Specifically, we provide a short proof of the second part of the conjecture, demonstrating that for any positive integers $b$ and $c$ with $2 \nmid b$ and $2 \mid c$, there exists an integer $n$ satisfying $1<n \le \max\{b, c\}$ such that $n^2 + bn + c$ is a practical number. Combined with the work of Somu, Li, and Kukla, this completely settles the conjecture of Wang and Sun.
We study the representation of even integers in the form $n=p+k^2$, where $p$ is a prime number and $k$ is an odd positive integer. After establishing the necessary modular constraints, we formulate a conjecture asserting that every sufficiently large such $n$ admits at least one such representation. We provide heuristic arguments based on the prime number theorem, numerical evidence up to $2 \times 10^6$, and discuss possible asymptotic formulas using the Hardy-Littlewood circle method. Several counterexamples are identified, and their structures are analyzed.
Moumouni Djassibo Woba, Kyelem Bila Adolphe, S. Attan et al.· JP Journal of Algebra Number...· 0 citations
We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $\Omega(ab)\leq33$, where $\Omega(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Friedlander--Iwaniec $\Lambda^-\Lambda^2$ lower-bound sieve to a sequence derived from the products $n(N-n)$. The main new ingredient is pre-sieving at the prime $3$, which eliminates the extremal small-prime case in the dimension condition while keeping the resulting remainder terms under explicit control. We complete the proof using analytic estimates for large $N$, finite verification over an intermediate range, and explicit prime-gap data for small $N$.
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.
Let $\ell(n)$ denote the least non-trivial reduced quadratic residue modulo $n$; that is, $\ell(n)$ denotes the smallest square-free integer $r>1$ with $(r,n)=1$ and $r\equiv x^2 \bmod {n}$. We establish nearly optimal bounds for $\ell(n)$, both in terms of the magnitude of $n$ and of its number of prime factors $\omega(n)$. In particular, we construct moduli $n$ for which $\ell(n)$ is unexpectedly large. As an application of our results, we prove bounds for the rate at which binary quadratic forms with bounded discriminant represent all positive integers up to $N$.
Kannan Soundararajan, João C. C. Vargas· 0 citations
Let $h$ be a positive integer, and let $A$ be a finite set of integers. We derive an exact formula for $|hA|$. Furthermore, let $A=\{0,1,\ldots,s,a,b\}$, $1\leq s<a<b$, and write $b=qa+r$ with $0\leq r<a$. By using generating function, we prove that $|hA|$ equals a definite explicit formula expressed in terms of certain truncated binomial coefficients for all positive integers $h$ if and only if $r=0$ or $qs+r\geq a$. This generalizes a result of Nathanson.
Let $C_n$ be a cyclic group of order $n$. We prove that if $(n,6)=1$, then every minimal zero-sum sequence of length four over $C_n$ has index one, thereby resolving the length-four index conjecture. After the gcd reduction, the nonunit case follows from the theorem of Shen-Xia-Li, and the remaining unit case is solved by a new multiplicative Fourier argument. The index-two residue identity yields a character-moment relation, and the odd characters with vanishing first moment form an exceptional spectrum of size at most $157\varphi(n)/1440<\varphi(n)/9$. A finite-group uncertainty principle then forces the four-term multiset to be invariant under negation, contradicting minimality. Apart from standard facts about primitive Dirichlet $L$-functions, the remaining argument is finite and requires neither asymptotic estimates nor computational verification.
Hongjian Li, Pingzhi Yuan, Shijie Yuan et al.· 0 citations
