Related papers: On a conjecture Mordell

In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$.…

In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}…

In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv3\pmod 4$ be a prime. Then $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{8^k}\equiv-\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{(-16)^k}\pmod{p^3}.$$ And for any odd prime…

We give a new proof of a theorem by P. Mihailescu which states that the equation $x^p-y^q=1$ is unsolvable with $x, y$ integral and $p, q$ odd primes, unless the congruences $p^q \equiv p\pmod{q^2}$ and $q^p\equiv q \pmod{p^2}$ hold.

In this paper, we mainly prove the following conjectures of Z.-H. Sun \cite{SH2}: Let $p>3$ be a prime. If $p\equiv1\pmod3$ and $p=x^2+3y^2$, then we have $$…

Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\ldots$ such that for sufficiently large primes $p$ we have $\sum_{k=0}^{p-1}a_k\equiv…

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the $7$-modular and $11$-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a…

In this note we provide some counterexamples for the conjecture of Moret\'{o} on finite simple groups, which says that any finite simple group $G$ can determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$…

We study a conjecture by Deaconescu on the solubility of finite groups with claims that if more than half of the elements in a finite group has the same order $k$, then the group is soluble. We show that the original conjecture fails by…

In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$.…

The polynomials $d_n(x)$ are defined by \begin{align*} d_n(x) &= \sum_{k=0}^n{n\choose k}{x\choose k}2^k. \end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \begin{align*}…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

We investigate the Lebesgue--Nagell equation \begin{align*} x^2-2=y^p \end{align*} in integers $x,y,p$ with $p\geq 3$ an odd prime. A longstanding folklore conjecture asserts that the only solutions are the ``trivial'' ones with $y=-1$. We…

In this paper, we prove that every prime $p$ which is congruent to $4,7$ modulo $9$ is the sum of two rational cubes. This is $2/3$ of Sylvester's conjecture which has a history of nearly 150 years since 1879. In the proof, we use recent…

Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture…

In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\equiv 1\pmod{4}$ or $a>1$ then $$…

A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p - 1)(q -…

Let $p$ be an odd prime and let $d\in\{2,3,7\}$. When $(\frac{-d}p)=1$ we can write $p=x^2+dy^2$ with $x,y\in\mathbb Z$; in this paper we aim at determining $x$ or $y$ modulo $p^2$. For example, when $p=x^2+3y^2$, we show that if $p\equiv…

Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset $E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral, meaning any complex-valued function $f$ on $E$ can be written as a linear…

The ternary Goldbach conjecture states that every odd number $m \geqslant 7$ can be written as the sum of three primes. We construct a set of primes $\mathbb{P}$ defined by an expanding system of admissible congruences such that almost all…