International Journal of Number Theory

Denote by $P^{+}(n)$ (respectively, $P^{-}(n)$) the largest (respectively, the smallest) prime factor of the integer $n\ge 1$. In this paper, we prove a lower bound of almost-primes $n\le x$ with $P^{-}(n)>x^{1/v},v>4$ such that $P^{+}(n-a)\le x^{1/u}$ for $1\le u\ll 1,a\in \mathbb{Z},a\ne 0$. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

We give a $q$-analogue of the supercongruence (I.2) of Van Hamme. As a conclusion, we confirm a recent conjecture of Swisher. We also give a $q$-analogue of the corresponding $\pi$ series (I.1) along with some similar results.

Using a $q$-series identity of Carlitz and a $q$-analogue of Morley’s congruence due to Pan, we give a proof of a $q$-congruence conjectured by Tauraso.

We describe an algorithm that allows to compute the Euclidean minimum (for the norm form) of any order of a totally definite quaternion field over a number field $K$ of degree strictly greater than $1$. Our approach is a generalization of the previous work dealing with number fields. The algorithm was practically implemented when $K$ has degree $2$.

In this paper, we primarily consider a generalization of the fermionic $p$-adic $q$-integral on ${\mathbb{Z}}_p$ including the parameters $\alpha$ and $\beta$ and investigate its some basic properties. By means of the foregoing integral, we introduce two generalizations of $q$-Changhee polynomials and numbers as $q$-Changhee polynomials and numbers with weight $(\alpha ,\beta )$ and $q$-Changhee polynomials and numbers of second kind with weight $(\alpha ,\beta )$. For the mentioned polynomials, we obtain new and interesting relationships and identities including symmetric relation, recurrence relations and correlations associated with the weighted $q$-Euler polynomials, $\lambda$-Stirling numbers of the second kind and Stirling numbers of first and second kinds. Then, we discover multifarious relationships among the two types of weighted $q$-Changhee polynomials and $p$-adic gamma function. Also, we compute the weighted fermionic $p$-adic $q$-integral of the derivative of $p$-adic gamma function. Moreover, we give a novel representation for the $p$-adic Euler constant by means of the weighted $q$-Changhee polynomials and numbers. We finally provide a quirky explicit formula for $p$-adic Euler constant.

Fix $l\ge 2$ and put $f(x)=(x+1)\cdot \dots \cdot (x+l)$. Improving on results of [M. Skałba, Products of disjoint blocks of consecutive integers which are powers, Colloq. Math. 98 (2003) 1–3], we prove the following. For a given $l\ge 2$ and each $𝜀>0$, there exists $C(𝜀)$ such that for any exponent $q$, there exist nonnegative integers $x_1,\dots ,x_k,y$ satisfying the equation ${\prod }_{i=1}^{k}f(x_i)=y^q$ and inequalities: $x_1<\cdots <x_k,x_{j+1}\ge x_j+l$ and

Let $K$ be a number field of degree $n$ over $\mathbb{Q}$ and $A$ its ring of integers. For a prime number $p\ge n$, we determine the types of splittings of $p$ in $K$ for which the set $\overline{J}=\{\overline{𝜃}\in A/pA,p|I(𝜃)\}$ is of cardinality a power of $p$. We prove that this necessary condition is also sufficient for $\overline{J}$ to be a subgroup of the additive group $A/pA$. Consequently, we show that, in this case, the subset of $A$, $J=\{𝜃\in A,p|I(𝜃)\}$ is an order of the number field.

Over the ring of algebraic integers ${𝒪}$ of a number field $K$, the quadratic congruence $a_1{x}_1^2+\cdots +a_k{x}_k^2\equiv c$ modulo a nonzero ideal ${ℐ}$ is considered. We prove explicit formulas for $N_k(a_1,\dots ,a_k,c,{ℐ})$ and $N_k^{\ast }(a_1,\dots ,a_k,c,{ℐ})$, the number of incongruent solutions $(x_1,\dots ,x_k)\in {{𝒪}}^k$ and the number of incongruent solutions $(x_1,\dots ,x_k)\in {{𝒪}}^k$ with $x_1{x}_2\cdots x_k$ coprime to ${ℐ}$, respectively. If ${ℐ}$ is contained in a prime ideal ${𝒫}$ containing the rational prime $p$, it is assumed that ${𝒫}$ is unramified over $p$. Moreover, some interesting identities for exponential sums are proved.

Let ${𝒮}$ be a commutative semigroup endowed with a binary associative operation $+$. An element $e$ of ${𝒮}$ is said to be idempotent if $e+e=e$. The Erdős–Burgess constant of ${𝒮}$ is defined as the smallest $\ell \in \mathbb{N}\cup \{\infty \}$ such that any sequence $T$ of terms from $S$ and of length $\ell$ contains a nonempty subsequence, the sum of whose terms is idempotent. Let $q$ be a prime power, and let ${𝔽}_q[x]$ be the polynomial ring over the finite field ${𝔽}_q$. Let $R={𝔽}_q[x]/K$ be a quotient ring of ${𝔽}_q[x]$ modulo any ideal $K$. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring $R$, in particular, we determined the Erdős–Burgess constant in the case when $K$ is the power of a prime ideal or a product of pairwise distinct prime ideals in ${𝔽}_q[x]$.

We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character $\left(\frac{-4}{\ast }\right)$, which are analogous to Ramanujan’s formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results.

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re (s)<1$. Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

In this paper, I generalize Holzer’s theorem for semi-diagonal equation.

In this paper, we prove a family of identities among multiple zeta values, which contains as a special case a conjectural identity of Hoffman. We use the iterated integrals on ${\mathbb{P}}^1\setminus \{0,1,\infty ,z\}$ for our proof.

Thanks to our recursive algorithm developed in [Trace of totally positive algebraic integers and integer transfinite diameter, Math. Comp. 78(266) (2009) 1119–1125], we prove that, if $\alpha$ is a totally positive algebraic integer of degree $d\ge 2$ with minimum conjugate ${\alpha }_1$ then, with a finite number of explicit exceptions,

On pages 51–53 of his lost notebook, S. Ramanujan has recorded several incomplete elliptic integral identities of the first kind. In the process, he raises a question “cases? for $\phi$ also”, where he intends to obtain identities involving the $\phi$ function. Motivated by this, we obtain two new identities involving the $\phi$ function.

In a recent paper, Sun posed six conjectures on the relations between $T(a_1,a_2,\dots ,a_k;n)$ and $N(a_1,a_2,\dots ,a_k;n)$, where $T(a_1,a_2,\dots ,a_k;n)$ denotes the number of representations of $n$ as $a_1\frac{x_1(x_1+1)}{2}+a_2\frac{x_2(x_2+1)}{2}+\cdots +a_k\frac{x_k(x_k+1)}{2}$, where $a_1,a_2,\dots ,a_k$ are positive integers, $n,x_1,x_2,\dots ,x_k$ are arbitrary nonnegative integers, and $N(a_1,a_2,\dots ,a_k;n)$ denotes the number of representations of $n$ as $a_1{x}_1^2+a_2{x}_2^2+\cdots +a_k{x}_k^2$, where this time $x_1,x_2,\dots ,x_k$ are integers. In this paper, we prove Sun’s six conjectures by using Ramanujan’s theta function identities.