Papers

Let $p_j$ be the prime with index $j$. We prove that the ratios $m/n$ for which $p_m+p_n$ is a square are dense in $\mathbb R_{>0}$. The same is true when $|p_m-p_n|$ is a square. In every nonempty open interval, almost every index can be used as a numerator and as a denominator. We also give a quantitative lower bound for the number of possible partners.

Let $\mathbb{N}_0=\{0,1,2,\ldots\}$ and let $h\ge2$ be an integer. For a set $A\subseteq\mathbb{N}_0$, write $hA$ for the set of all sums of $h$, not necessarily distinct, elements of $A$. In this paper, we prove that for every $h\ge2$, there is a partition $\mathbb{N}_0=A\sqcup B$ such that $A$ is a minimal asymptotic basis of order $h$ and $B$ is a maximal asymptotic nonbasis of order $h$. This solves an open problem posed by Nathanson in 1974.

For any integers $x$ and $y$, let $(x,y)$ and $[x,y]$ stand for the greatest common divisor and the least common multiple of $x$ and $y$, respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1,...,x_n\}$ be a set of $n$ distinct positive integers. Let $(f((x_i,x_j)))$ (abbreviated by $f(S)$) and $(f([x_i,x_j]))$ (abbreviated by $(f([S]))$) stand for the $n\times n$ matrices whose $(i,j)-$entry is $(f((x_i,x_j)))$ and $(f([x_i,x_j]))$ respectively. In 1989, Beslin and Ligh gave a description of the lower triangular decomposition of $((x_i,x_j))$. In 1992, Bourque and Ligh showed that if $S$ is factor closed (i.e., S contains all positive divisors of any element of S), then the GCD matrix $((x_i,x_j))$ divides the LCM matrix $([x_i,x_j])$ (written as $((x_i,x_j))|([x_i,x_j])$) in the ring $M_n(\mathbb{Z})$ of $n\times n$ matrices over the integers. In this paper, we will show the diagonalization of $((x_i,x_j))$ and its applications. Our main new contributions are Theorems 4.1 and 4.2, which extend previous results to gcd-closed sets satisfying condition $\mathcal{G}$.

Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{\gamma_k+o(1)}$ for a well-defined exponent $\gamma_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $\gamma_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ \gamma_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang–Zhang bounds, and gives $0.438899\ldots<\gamma_3\le0.889881\ldots$.

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in $\mathbb{F}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.

We establish five families of integer-parameter extensions of Ramanujan's partial theta identities. The families follow from a common two-parameter specialization of Andrews' transformation, for which we give an independent proof based on a $q$-difference recurrence and a boundary estimate. Specializations of the integer parameter recover six identities from Ramanujan's lost notebook. As an application, a residue argument applied to the fifth family gives an integer-parameter extension of Lovejoy's residual identity, from which we construct a corresponding family of conjugate Bailey pairs.

Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)t^n/n! = 1+\sum_{m\ge1}\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\varphi(k)$. We prove the stronger statement that the Carmichael function $\lambda(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\equiv a(np^{r-1}+i)\pmod{p^r}$ for every $i\ge0$, every prime $p$, and all $n,r\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\in\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.

We answer a question of C. S. Herz about the number of integral $k\times m$ matrices $T$ such that $T^tT\leq R$ for a fixed positive definite $m\times m$ matrix $R$. We give an asymptotic formula for the count. This can be seen as a generalization of the Gauss circle problem, as counting representation of quadratic forms by the identity $I_k$, and as counting integral points bounded by the Stiefel manifold $T^tT=R$. The main tool is a bound for Bessel functions of matrix argument that was proved over Jordan algebras.

We give an approach to discover continued fractions for series of the form $$\sum_{k=0}^\infty \frac{\epsilon^k}{(x+k)^s},$$ where $\epsilon = \pm 1$. We find a continued fraction of the form \begin{equation*} \frac{a_1}{b_1(x)} \fplus \frac{a_2}{b_2(x)} \fplus \fdots \end{equation*} where $a_k$ are constants and $b_k(x)$ are polynomials. Our technique involves telescoping continued fractions. This provides a discovery approach to continued fractions given by Ramanujan for $$2\sum_{k=1}^\infty \frac{(-1)^{k+1}}{x+2k-1}, 2\sum_{k=0}^\infty \frac{1}{(x+2k+1)^2}, 2\sum_{k=0}^\infty \frac{(-1)^k}{(x+2k+1)^2}, \sum_{k=1}^\infty \frac{1}{(x+k)^3}, $$ and the like. We display the first few terms of several continued fractions obtained in this manner for larger values of $s$, including $s=5, 7, 9, 11$. They do not follow as simple a pattern as Ramanujan's continued fractions.

We establish logarithmic upper bounds for nonnegative multiplicative functions evaluated at values of multivariable polynomials. The polynomial contribution is encoded by the permutation character on the geometric irreducible components of the corresponding hypersurface, while the arithmetic contribution is described by a class function on a joint Chebotarev–Sato–Tate group. This yields a unified framework for symmetric-power coefficients of CM and non-CM modular forms, Dedekind zeta-function coefficients, and mixed automorphic–Galois weights. As a further application, we obtain quantitative divisibility results for Fourier coefficients along polynomial values, including explicit formulas in the full residual-image and Eisenstein congruence cases.

Let $p \geq 5$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E/K$ be an elliptic curve with good reduction. Let $F_n$ denote the $n$-division polynomial of $E$. Silverman proved that if the reduction is ordinary, then for every $P \in E(K) \setminus \hat{E}(K)$ and a suitable power $q$ of $p$, the sequences $(F_{mq^k}(P))_{k \geq 0}$ converge $p$-adically to limits that are algebraic over the field of definition of $E$. In both the ordinary and supersingular cases, we show that these sequences converge, and determine these limits explicitly in terms of the values of Mumford's algebraic sigma function attached to the Teichmüller lift of the prime-to-$p$ torsion component of the reduction of $P$. In particular, the limits are algebraic in the supersingular case as well. As an application, we obtain explicit $p$-adic limit formulas for nonsingular elliptic divisibility sequences.

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $\gamma_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $\gamma_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le \gamma_{1,1,-3}\le 1/2-\delta$ for some constant $\delta>0$. In addition we show that the limit as $a \rightarrow \infty$ of $\gamma_{1,1,-a}$ is equal to precisely $1/5$.

We prove the irrationality of \[ \Li_2(-1/4),\qquad \Li_2(1/5),\qquad \Li_2(-1/3). \] The three arguments form a natural progression. The first is an endpoint completion of the five-parameter Rhin–Viola method. Rhin and Viola's 2019 treatment already supplies the negative-argument continuation, permutation invariance and factorial divisor; at $z=-4$ we use three bounded shifts, an elementary fixed-contour estimate and a four-term recurrence to remove the remaining nonvanishing problem in the complex-saddle regime. For $1/5$ we pass to the six-parameter Viola–Zudilin family and apply the Rhin–Viola factorial transformations term by term inside two binomial expansions of the same integral. This inherited divisor crosses the arithmetic threshold that the unrefined five-parameter construction does not reach in our computations. The same crossbreed, combined with the negative-argument continuation, proves the result at $-1/3$. All proof-critical finite inequalities are accompanied by exact executable certificates. Supplementary computations propagate $10{,}000$ exact primitive coefficient pairs for each of $1/5$ and $-1/3$, with complete gcd removal and no failure of the certified smallness or adjacent nonproportionality checks.

In this paper, we prove Hare's $\epsilon$-unfair conjecture in [Computational progress on the unfair 0–1 polynomial conjecture, Experiment. Math. (2025), https://doi.org/10.1080/10586458.2025.2527786].

Let $F$ be a function field over an algebraically closed field $K$ and $S$ a finite dimensional $K$-subspace of $F$. The square of $S$ is spanned by all products of pairs of elements in $S$. We conjecture that if $\dim S^2 \leq 3 \dim S - 4$, then $S^2$ must contain a Riemann–Roch space of dimension at least $2 \dim S - 1 + g$, where $g$ is the genus of $F$. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that $S$ is contained in a Riemann–Roch space of dimension at most $3/2\dim S + 1$. For the proof we study the annihilator of $S$ and introduce the notion of weight for linear forms.

We prove an explicit Fricke transformation for the two-dimensional space generated by normalized reciprocal products on the cubic-residue cosets modulo 13. The transformation matrix is a scalar multiple of a matrix of differences of cubic Gaussian periods. Its projective action is defined over the cyclic cubic field, whereas the normalized matrix is defined over the real cyclotomic field. The proof uses generalized eta functions, a modular unit of degree two on $X_1(13)$, and a five-coefficient identity. A separate divisor argument lifts the projective transformation to the asserted linear transformation. We also record the Fricke images at an arbitrary prime $p\equiv1\pmod3$ and prove that their normalized sine constants belong to the associated cyclic cubic subfield.

We generalize Zeckendorf's theorem to second-order linear recurrences of the form $G_{k+2} = gG_{k+1} + hG_{k}$ with arbitrary, coprime initial values $(G_1, G_2)$. We analyze the asymptotic behavior of the count $\#R_G(X)$ of positive integers less than or equal to $X$ that have an expansion under the standard rule of expansions $\mathcal{E}$ associated with the recurrence. Additionally, we provide sufficient conditions under which $G$ has the unique expansion property under the rule of expansions. Finally, we completely characterize the $g$-golden ratio recurrence family when $G_1 = 1$ or $G_2 = 1$. Our approach directly investigates the algebraic structure of the expansions, simplifying and completing previous partial results.

We establish a uniform product-difference theorem for dense subsets of $\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta>0$ there exists an integer $K(\delta)\geq 1$ such that every set $E\subseteq\mathbb Z^2$ with upper Banach density $d^\star(E)\geq\delta$ satisfies \[ K(\delta)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq\mathbb Z$ and $E\subseteq\mathbb Z^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\mathbb Z^2$.

Cyclotomic divisibility imposes strong prime-power structure on integer tiles. We study unsupported cyclotomic divisors: mixed-order divisors for which none of the prime-power components of the order divides the mask, although every prime in the order divides the tile cardinality. Kiss, Łaba, Marshall and Somlai asked whether such a phenomenon can occur in the three-prime setting. We prove that unsupported cyclotomic divisors already occur for periods with three distinct prime factors. For primes \(p<q<r\), we characterize the square-period case: an unsupported factor \(\Phi_{pqr}\) occurs in a tiling of \(\ZZ_{(pqr)^2}\) if and only if \(r\in\langle p,q\rangle\), and every such tile lies in a single residue class modulo \(r\). Among cyclic tilings with the unsupported order dividing the specified modulus, the smallest modulus is \(180\); if the order has three distinct prime factors, it is \(900\). An Apéry-set construction gives examples for every triple at period \(p^2q^2r^3\).The proof of our results combines Fourier rigidity, a three-cylinder decomposition, and an integer mass obstruction.

For a constructible sheaf $\mathcal{F}$ on a smooth scheme $X$ over a perfect field, the singular support $SS(\mathcal{F})$ and the characteristic cycle $CC(\mathcal{F})$ are defined on the cotangent bundle. In analogy with the construction of Kashiwara–Schapira via microlocalization in the transcendental setting, we construct the microlocal versions ${ SS}_\mu(\mathcal{F})$ and $CC_\mu(\mathcal{F})$ of $SS(\mathcal{F})$ and $CC(\mathcal{F})$ respectively, and prove \[ SS_\mu(\mathcal{F})= SS(\mathcal{F}),\qquad CC_\mu(\mathcal{F})= {\rm cl}CC(\mathcal{F}), \] where $\operatorname{cl}$ is the cycle class map. As a consequence of the second equality, we show that the cohomological characteristic classes are the cycle classes associated with the corresponding characteristic cycles, as conjectured in (Invent.~Math.~207: 597–695, 2017). As further applications, we give new proofs of the index formula and Bloch's conductor formula in the proper case, independently of Abe's homotopical approach via $\infty$-categories.

The quadratic sieve method is a core tool in number theory. In this paper, we present two optimization methods for the Quadratic Sieve algorithm. In the sieving process, the original polynomial root value accumulation step is changed from the original di to the mdi (m is a small integer), which can change the complexity from O(n) to O(n/m). Another optimization method is that for all parameter lookup steps, the original traversal lookup can be changed to an efficient binary search, which can change the complexity of the loop from O(n) to O(logn). This enhancement reduces the computational complexity of RSA modulus factorization in practical settings.

As a valuable theoretical and application problem, the integer points on two types of elliptic curve y^2=(x-a)(x-b)(x-c) and y^2 = (x-a)(x^2+ax+b) are studied. By using elementary number theory method, the solution of the original equations is reduced to the solution of simultaneous Pell equations or generalized Pell equation, which can improve the efficiency in searching integer solutions by computer. Effective methods are proposed to find the integer solutions of the equations, which are convenient for algorithm programming implementation. At the same time, the derivation of the methods leads to two sufficient conditions, which can determine the unsolvability of these two types of equation respectively. The methods proposed are for equations with undetermined coefficients, which are more general than the solutions of the equations with specific known coefficients, and more suitable for computer programming implementation.

Let $E$ be an elliptic curve over $\mathbb{Q}_p$. We study the field $\mathbb{Q}_p(E[p])$ generated by the $p$-torsion points of $E$. When $E$ has good reduction, we determine not only $\mathbb{Q}_p(E[p])$ but also the field $\mathbb{Q}_p(P)$ for every point $P\in E[p]$. This classification yields criteria for the existence of $p$-torsion over unramified and Lubin–Tate extensions of $\mathbb{Q}_p$. As a global application, let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction. Define $f_p(E)$ to be the multiplicative order of $a_p(E)$ modulo $p$ in the ordinary case, and set $f_p(E)=p^2-1$ in the supersingular case. We prove that $E(K)[p]=0$ for every number field $K/\mathbb{Q}$ with $[K:\mathbb{Q}]<f_p(E)$. When $E$ has bad reduction, we describe the ramified part of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$. For every reduction type, we determine the maximal upper ramification break of $\mathbb{Q}_p(E[p])/\mathbb{Q}_p$.

We prove a new estimate on Jacquet-Whittaker function for $\mathrm{GL}_n(\mathbb R)$, assuming that the Langlands parameters are purely imaginary and well-spaced. This gives an upper bound for the global sup-norm of the Whittaker function that matches the lower bound of Brumley-Templier in the exponent up to a linear error in $n$.

We consider the family of cyclic fourfold covers $w^4 = \prod_{j=1}^{7}(z-x_j)$ of the complex projective line branched at the eight points $x_1,\ldots,x_7,\infty$. The period map identifies the configuration space $X(2,8)$ of the branch points with a Zariski open subset of a quotient of the five-dimensional complex ball. The inverse of the period map is expressed projectively by $105$ automorphic forms $f_J$, which are proportional to the signed branch-point polynomials $\hat x_J$ with a common scalar factor. We determine this factor for the period $\eta$ of the differential $dz/w$: it is the product of the constant $-1/(2^{12}\Gamma(3/4)^{16})$ and the square of a quadratic form in $\eta$.

We obtain asymptotic formulas for the three sums \[ \sum_{n\le x}\frac{\psi(\psi(n))}{\psi(n)},\qquad \sum_{n\le x}\frac{\psi(n)}{\psi(\psi(n))},\qquad \sum_{n\le x}\log\frac{\psi(\psi(n))}{\psi(n)}, \] where \(\psi\) denotes Dedekind's arithmetic function. We also determine the normal order of the successive-iterate ratio; more precisely, \[ \frac{\psi(\psi(n))}{\psi(n)} \sim \frac{6e^\gamma}{\pi^2}\log_3 n \] for almost all positive integers \(n\).

We study the iteration of (T(n)=\psi(n)-n), where (\psi) is the Dedekind psi function. We show that, for almost all integers, successive iterates satisfy strong asymptotic growth relations governed by (T(n)/n). As an application, we prove that for every fixed (\ell\ge1), the set of integers satisfying (T^{(\ell)}(n)=n) has asymptotic density zero, and hence so does the set of periodic points of any fixed period.

We prove that every nonzero integer polynomial with at least two distinct complex roots has lattice point visibility density one, resolving the generalized form of the Visibility Density Conjecture for nonzero polynomials. This extends the origin-passing case established by Chaubey, Pandey, and Regavim. Visibility is taken along the curves $y=tF(x)$ with rational $t$, with a point visible if no positive lattice point on the same curve has a smaller horizontal coordinate. The proof uses a greatest common divisor cutoff to reduce the problem to finitely many equations of the form $F(b)=qF(a)$, with $0<q<1$; for each equation, the positive integers $a$ admitting a positive integer solution $b<a$ form a set of density zero. This elementary argument removes the proper-power hypothesis of earlier work without requiring estimates uniform in the ratio.

We prove the Tate conjecture for abelian fivefolds over finite fields. The proof constructs correspondences for a residual motive using a Moret–Bailly family, Gross–Schoen heights, and monodromy. We also prove standard conjecture~$D_\ell$ over $\overline{\mathbf F}_p$ and independence of $\ell$ of rational cycle class kernels over algebraically closed fields of characteristic $p$. Over finite fields, rational and numerical equivalence agree with rational coefficients, and higher algebraic $K$-groups vanish rationally.

We prove that the number of curves of a fixed genus g with n marked points over finite fields is a polynomial function of the cardinality of the field if and only if g = 0 or 3g + 2n is less than or equal to 24. This confirms a conjecture of Canning, Larson, and the authors.