In this paper, we investigate sums of four squares of integers whose prime factorizations are restricted, making progress towards a conjecture of Sun that states that two of the integers may be restricted to the forms $2^a3^b$ and $2^c5^d$. We obtain an ineffective generalization of results of Gauss and Legendre on sums of three squares and an effective generalization of Lagrange's four-square theorem.
In this paper, we will give another proof of Zhi-Wei Sun's three conjectures on Apéry-like sums involving harmonic numbers by proving some identities among special values of multiple polylogarithms.
In this paper, we show that certain sums of generalized $m$-gonal numbers represent every positive integer if and only if they represent every positive integer up to an explicit bound $C_m$, verifying a conjecture of Sun for sufficiently large positive integers.
In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.
On the minimum size of restricted sumsets in cyclic groups
对于正整数$n$、$m$和$h$，我们令$ρ\hat{\;}(\mathbb{Z}_n, m, h)$表示$n$阶循环群的所有$m$ -子集中$h$ -倍限制和集的最小大小. $ρ\hat{\;}(\mathbb{Z}_n, m, h)$的值在1960年代由Erdens和Heilbronn对$n$和$h=2$的素数值进行了证明; Dias da Silva和Hamidoune在1994年证明了该猜想，并将其推广到任意的$h$，但对$n$是合成的情况知之甚少。在这里，我们展示了所有$n$、$m$和$h$的显式上界;我们的界对于所有已知情况（包括所有$n$、$m$和$h$与$n \leq 40$）都是紧的。我们还提供了由Plagne和Hamidoune，Lladó和Serra所作的解释的反例。
Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.
A sequence $\bfx=(x_1,\ldots,x_m)$ of elements of $\Z_n$ is called an \textit{$A$-weighted Davenport Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\bfzero_m$ such that $\sum_i a_ix_i=0$. Here $\bfzero_m=(0,\ldots,0)\in\Z_n^m$. Similarly, the sequence $\bfx$ is called an \textit{$A$-weighted Erdős Z-sequence} if there exists $\bfa:=(a_1,\ldots,a_m)\in (A\cup\{0\})^m\setminus\{\bfzero_m\}$ with $|Supp(\bfa)|=n$, such that $\sum_i a_ix_i=0$, where $Supp(\bfa):=\{i: a_i\ne 0\}$. A $\Z_n$-sequence $\bfx$ is called $k$-restricted if no element of $\Z_n$ appears more t…
Let $k\geqslant 3$ and let $A=\{0=a_{0}<a_{1}<\cdots<a_{k-1}\}$ with $\gcd(A)=1$. Freiman-Lev conjecture [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40] is a well-known conjecture which related to restricted sumsets. Up to now, Freiman-Lev conjecture is open for all $a_{k-1}\geqslant 2k-2$. In this paper, we prove the Freiman-Lev conjecture is true for $a_{k-1}\geqslant 2k-2$ and $a_{k-2}<2k-4$. That is, Freiman-Lev conjecture is still open for the case $a_{k-1}\geqslant 2k-2$ and $a_{k-2}\geq 2k-4$.
In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.
We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.
The restricted partition function $p_{N}(n)$ counts the partitions of $n$ into at most $N$ parts. In the nineteenth century Sylvester showed that these partitions can be expressed as a sum of $k$-periodic quasi-polynomials ($1\leq k\leq N$) which he termed as Waves. It is now well-known that one can easily perform a wave decomposition using a special type of partial fraction decomposition (the so-called $q$-partial fractions) of the generating function of $p_{N}(n)$. In this paper we show that the coefficients of these $q$-partial fractions can be expressed as a linear combination of the Raman…
Proof of Sun's conjectures on integer-valued polynomials
最近，Z W. Sun引入了两类与Delannoy数有关的多项式，并证明了涉及这两类多项式的和的超同余。我们推导出新的求和公式平方的多项式，并使用它们来证明，某些合理的总和，这些多项式的权力，甚至是整数，每当他们被评估为整数。这证实了Z的两个版本。W.太阳我们还推测，许多这些结果有整洁的$q$ -类似物。
In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also conjectured that for each positive integer $n$ there exist integers $x,y,z,w$ such that $n=x^2+y^2+z^2+w^2$ and $x+3y\in\{4^k:k\in\mathbb{N}\}$. In this paper, we confirm these conjectures via some arithmetic theory of ternary quadratic forms.
