Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E:y^2=x^3-4p^{2}x$. Further, let $N\left(p\right)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\ge 17$, then $N\left(p\right)\le 4$ or $1$ depending on whether $p\equiv 1(mod8)$ or $p\equiv -1(mod8)$.

By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$...

Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m,d}:=\mathbb{Q}({\zeta }_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m,d}$ having an odd class number. Furthermore, using the cyclotomic ${\mathbb{Z}}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m,d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5(mod8)$.

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and $U_{n+1}=PUₙ-Q{U}_{n-1}$ for n ≥ 1, and V₀ = 2, V₁ = P and $V_{n+1}=PVₙ-Q{V}_{n-1}$ for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and $V_r\ne 1$. We show that there is no integer x such that $Vₙ=V_{r}Vₘx²$ when m ≥ 1 and r is an even integer. Also we completely solve the equation $Vₙ=VₘV_{r}x²$ for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Let $K$ be a number field. We consider a local-global principle for elliptic curves $E/K$ that admit (or do not admit) a rational isogeny of prime degree $\ell$. For suitable $K$ (including $K=\mathbb{Q}$), we prove that this principle holds for all $\ell \equiv 1mod4$, and for $\ell \<7$, but find a counterexample when $\ell =7$ for an elliptic curve with $j$-invariant $2268945/128$. For $K=\mathbb{Q}$ we show that, up to isomorphism, this is the only counterexample.

Let $K=\mathbb{Q}\left(\alpha \right)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f\left(x\right)=x^{18}-m$, $m\ne \mp 1$, is a square free rational integer. We prove that if $m\equiv 2$ or $3(\mathrm{mod}4)$ and $m\neg \equiv \mp 1(\mathrm{mod}9)$, then the number field $K$ is monogenic. If $m\equiv 1(\mathrm{mod}4)$ or $m\equiv 1(\mathrm{mod}9)$, then the number field $K$ is not monogenic.

Let $X\subset {\mathbb{P}}^n$ be an integral and non-degenerate $m$-dimensional variety defined over $ℝ$. For any $P\in {\mathbb{P}}^n\left(ℝ\right)$ the real $X$-rank $r_{X,ℝ}\left(P\right)$ is the minimal cardinality of $S\subset X\left(ℝ\right)$ such that $P\in \langle S\rangle$. Here we extend to the real case an upper bound for the $X$-rank due to Landsberg and Teitler.

Let ${\Phi }_1,...,{\Phi }_{k+1}$ (with $k\ge 1$) be vector fields of class $C^k$ in an open set $U{\subset }^{N+m}$, let $𝕄$ be a $N$-dimensional $C^k$ submanifold of $U$ and define $$𝕋:=\{z\in 𝕄:{\Phi }_1\left(z\right),...,{\Phi }_{k+1}\left(z\right)\in T_z𝕄\}$$ where $T_z𝕄$ is the tangent space to $𝕄$ at $z$. Then we expect the following property, which is obvious in the special case when $z_0$ is an interior point (relative to $𝕄$) of $𝕋$: If $z_0\in 𝕄$ is a $(N+k)$-density point (relative to $𝕄$) of $𝕋$ then all the iterated Lie brackets of order less or equal to $k$ $${\Phi }_{i_1}\left(z_0\right),[{\Phi }_{i_1},{\Phi }_{i_2}]\left(z_0\right),[[{\Phi }_{i_1},{\Phi }_{i_2}],{\Phi }_{i_3}]\left(z_0\right),...(h,i_h\le k+1)$$ belong to $T_{z_0}𝕄$. Such a property has been proved in [9] for $k=1$ and its proof in the...

Let ${\Phi }_n\left(q\right)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1(mod4)$, $$\sum _{k=0}^{n-1}\frac{{(q^{-1};q^2)}_k^2{(q^{-2};q^4)}_k}{{(q^2;q^2)}_k^2{(q^4;q^4)}_k}{q}^{6k}\equiv 0(mod{\Phi }_n{\left(q\right)}^2),$$ where ${(a;q)}_m=(1-a)(1-aq)\cdots (1-a{q}^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus ${\Phi }_n{\left(q\right)}^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo ${\Phi }_n{\left(q\right)}^2$ for $n\equiv 3(mod4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a ${}_4{\varphi }_3$ summation formula.

A space $X$ is said to have the Rothberger property (or simply $X$ is Rothberger) if for every sequence $\langle {𝒰}_n:n\in \omega \rangle$ of open covers of $X$, there exists $U_n\in {𝒰}_n$ for each $n\in \omega$ such that $X={\bigcup }_{n\in \omega }{U}_n$. For any $n\in \omega$, necessary and sufficient conditions are obtained for $C_p{(\Psi \left(𝒜\right),2)}^n$ to have the Rothberger property when $𝒜$ is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family $𝒜$ for which the space $C_p{(\Psi \left(𝒜\right),2)}^n$ is Rothberger for all $n\in \omega$.

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.