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 ${\Phi }_n\left(q\right)$ 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.

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 $5(mod8)$.

A positive integer $n$ is called a square-free number if it is not divisible by a perfect square except $1$. Let $p$ be an odd prime. For $n$ with $(n,p)=1$, the smallest positive integer $f$ such that $n^f\equiv 1(modp)$ is called the exponent of $n$ modulo $p$. If the exponent of $n$ modulo $p$ is $p-1$, then $n$ is called a primitive root mod $p$. Let $A\left(n\right)$ be the characteristic function of the square-free primitive roots modulo $p$. In this paper we study the distribution $$\sum _{n\le x}A\left(n\right)A(n+1),$$ and give an asymptotic formula by using properties of character...

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 $p\equiv -1(mod8)$.

Let $C_m:y^2=x^3-m^{2}x+p^2{q}^2$ be a family of elliptic curves over $\mathbb{Q}$, where $m$ is a positive integer and $p$, $q$ are distinct odd primes. We study the torsion part and the rank of $C_m\left(\mathbb{Q}\right)$. More specifically, we prove that the torsion subgroup of $C_m\left(\mathbb{Q}\right)$ is trivial and the $\mathbb{Q}$-rank of this family is at least 2, whenever $m\neg \equiv 0(mod3)$, $m\neg \equiv 0(mod4)$ and $m\equiv 2(mod64)$ with neither $p$ nor $q$ dividing $m$.

Let $K=\mathbb{Q}\left(\alpha \right)$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F\left(x\right)=x^{2^r·3^k·7^s}-m\in \mathbb{Z}\left[x\right]$, where $r$, $k$, $s$ are three positive natural integers. The purpose of this paper is to study the monogenity of $K$. Our results are illustrated by some examples.

For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive integer solution of the Pell equation $u^2-D{v}^2=1,$ and let $h\left(4D\right)$ denote the class number of binary quadratic primitive forms of discriminant $4D$. If $D$ satisfies $2\nmid D$ and $v_{1}h\left(4D\right)\equiv 0(modD)$, then $D$ is called a singular number. In this paper, we prove that if $(x,y,z)$ is a positive integer solution of the equation $x^y+y^x=z^z$ with $2\mid z$, then maximum $max\{x,y,z\}<480000$ and both $x$, $y$ are singular numbers. Thus, one can possibly prove that the equation has no positive integer solutions...

Let $E$ and $f$ be an Eisenstein series and a cusp form, respectively, of the same weight $k\ge 2$ and of the same level $N$, both eigenfunctions of the Hecke operators, and both normalized so that $a_1\left(f\right)=a_1\left(E\right)=1$. The main result we prove is that when $E$ and $f$ are congruent mod a prime $𝔭$ (which we take in this paper to be a prime of $\overline{\mathbb{Q}}$ lying over a rational prime $p\&gt;2$), the algebraic parts of the special values $L(E,\chi ,j)$ and $L(f,\chi ,j)$ satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n\in S_{k+1/2}(N,{\chi }_0)$ be a nonzero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus ${\chi }_0$, where the Fourier coefficients $a\left(n\right)$ are real. Bruinier and Kohnen conjectured that the signs of $a\left(n\right)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies ${\left\{a\left(t{n}^2\right)\right\}}_n$, where $t$ is a squarefree integer such that $a\left(t\right)\ne 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that ${\left\{a\left(t{n}^2\right)\right\}}_n$ is equidistributed over any arithmetic progression $n\equiv dmodq$.