On consecutive happy numbers

快乐函数的一种树方法

A tree approach to the happy function

本文通过树结构刻画 $e$ 次幂 $b$-快乐数的高度与祖先关系，给出单次迭代原像基数公式，并将方法推广到非快乐数。

On the Density of Happy Numbers

The happy function $H: \mathbb{N} \rightarrow \mathbb{N}$ sends a positive integer to the sum of the squares of its digits. A number $x$ is said to be happy if the sequence $\{H^n(x)\}^\infty_{n=1}$ eventually reaches one. A basic open question regarding happy numbers is what bounds on the density can be proved. This paper uses probabilistic methods to reduce this problem to experimentally finding suitably large intervals containing a high (or low) density of happy numbers as a subset. Specifically we show that $\bar{d} > .18577$ and $\underline{d} < .1138$. We also prove that the asymptotic d…

Repdigits as products of consecutive balancing or Lucas-balancing numbers

Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers.

On the distribution of consecutive square-free numbers of the form $\mathbf{[αn], [αn]+1}$

In the present paper we show that there exist infinitely many consecutive square-free numbers of the form $[αn]$, $[αn]+1$, where $α>1$ is irrational number with bounded partial quotient or irrational algebraic number.

Numbers and the Heights of their Happiness

A generalized happy function, $S_{e,b}$ maps a positive integer to the sum of its base $b$ digits raised to the $e^\text{th}$ power. We say that $x$ is a base $b$, $e$ power, height $h$, $u$ attracted number if $h$ is the smallest positive integer so that $S^{h}_{e,b}(x)=u$. Happy numbers are then base 10, 2 power, 1 attracted numbers of any height. Let $σ_{h,e,b}(u)$ denote the smallest height $h$, $u$ attracted number for a fixed base $b$ and exponent $e$ and let $g(e)$ denote the smallest number so that every integer can be written as $x_{1}^{e}+x_{2}^{e}+...+x_{g(e)}^{e}$ for some nonnegat…

Consecutive primes which are widely digitally delicate and Brier numbers

Making use of covering systems and a theorem of D. Shiu, the first and second authors showed that for every positive integer $k$, there exist $k$ consecutive widely digitally delicate primes. They also noted that for every positive integer $k$, there exist $k$ consecutive primes which are Brier numbers. We show that for every positive integer $k$, there exist $k$ consecutive primes that are both widely digitally delicate and Brier numbers.

Consecutive square-free numbers of a special form

In the present paper we prove that for any fixed $1<c<7/6$ there exist infinitely many consecutive square-free numbers of the form $[n^c], [n^c]+1$ and we also establish an asymptotic formula in given interval.

Product of Two Consecutive Fibonacci or Lucas Numbers Divisible by their Prime Sum of Indices

We show that the product of two consecutive Fibonacci (respectively Lucas) numbers is divisible by the sum of their indices if this sum is a prime number different from 5 and in the form (4r+1)(respectively (4r+3)).

Large gaps between consecutive prime numbers containing square-free numbers and perfect powers of prime numbers

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

The Differences Between Consecutive Smooth Numbers

We show that large gaps between smooth numbers are infrequent. The key new tool is a novel mean value bound for a special type of Dirichlet polynomial.

Consecutive positive integers with the same number of divisors

We establish new upper bounds for the length of runs of consecutive positive integers each with exactly $k$ divisors, where $k$ is a given positive integer of some special forms. Also we have found exact values of the maximum possible runs for some fixed values of $k$. In addition, we exhibit the run of 11 consecutive positive integers each with exactly 36 divisors, the run of 13 consecutive positive integers each with exactly 12 divisors, the run of 17 consecutive positive integers each with exactly 24 divisors, and the run of 17 consecutive positive integers each with exactly 48 divisors.

Large gaps between consecutive prime numbers containing perfect powers

For any positive integer $k$, we show that infinitely often, perfect $k$-th powers appear inside very long gaps between consecutive prime numbers, that is, gaps of size $$ c_k \frac{\log p \log_2 p \log_4 p}{(\log_3 p)^2}, $$ where $p$ is the smaller of the two primes.

On the differences between consecutive prime numbers, I

We show by an inclusion-exclusion argument that the prime $k$-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime $k$-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.

存在按判别式排序的连续三次域具有大类数

There are consecutive cubic fields with large class numbers, when ordered by discriminant

研究显示，按判别式排序的三次域中可存在任意长的序列，其中所有域的类数均超过给定界限。

On the Difference between Consecutive Primes and Estimates of the Number of Primes in the Interval $(n, 2n)$

Using evaluations of the difference between consecutive primes we develop another way of estimating of the number of primes in the interval $(n, 2n)$. We also discuss the ultra Cramer conjecture, $p_{n+1} - p_n = O(log^{1+ε}p_n)$ where $ε> 0$, in the context of the results we have obtained in our paper.

三个连续的幂数等差数列

Three-term arithmetic progressions of consecutive powerful numbers

研究证明存在无限多个等差数列，其中公差d=2√N+1，且探讨了这些数列是否为连续幂数的猜想。

On differences between consecutive numbers coprime to primorials

We consider the ordered sequence of coprimes to a given primorial number and investigate differences between consecutive elements. The Jacobsthal function applied to the concerning primorial turns out to represent the greatest of these differences. We will explore the smallest even number which does not occur as such a difference. Little is known about even natural numbers below the respective Jacobsthal function which cannot be represented as a difference between consecutive numbers coprime to a primorial. Existence and frequency of these numbers have not yet been clarified. Using the relatio…

On consecutive quadratic non-residues: a conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less than the square root of p. This paper uses elementary methods to prove that 13 is the only prime number for which the greatest number of consecutive quadratic non-residues modulo p exceeds the square root of p.

Large gaps between consecutive prime numbers containing perfect $k$-th powers of prime numbers

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a fixed positive constant, for which the interval $(p_n, p_{n+1})$ contains the $k$-th power of a prime number.

关于三个连续幂数的注记

A note on three consecutive powerful numbers

本文通过佩尔方程、椭圆曲线和二阶递推，证明了不存在中间项为完全立方数且两端各仅含一个奇次幂质因数的三个连续幂数三元组。

On Proving of Diophantine Inequalities with Prime Numbers by Evaluations of the Difference between Consecutive Primes

Using as the working hypothesis of an evaluation of the difference between primes $p_{n+1} - p_n = O(\sqrt{p_n})$ we represent in detail the proofs of Legendre's and Oppermann's conjectures.

On the constant in Burgess' bound for the number of consecutive residues or non-residues

We give an explicit version of a result due to D. Burgess. Let $χ$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $χ$ takes on a particular value is less than $\left\{\frac{πe\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ term is given explicitly.

On the alternating sums of powers of consecutive q-integers

In this paper we construct $q$-Genocchi numbers and polynomials. By using these numbers and polynomials, we investigate the $q$-analogue of alternating sums of powers of consecutive integers due to Euler.

Difference in the Number of Summands in the Zeckendorf Partitions of Consecutive Integers

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. We study the difference between the number of summands in the partition of two consecutive integers. In particular, let $L(n)$ be the number of summands in the partition of $n$. We characterize all positive integers such that $L(n) > L(n+1)$, $L(n) < L(n+1)$, and $L(n) = L(n+1)$. Furthermore, we call $n+1$ a peak of $L$ if $L(n) < L(n+1) > L(n+2)$ and a divot of $L$ if $L(n) > L(n+1) < L(n+2)$. We characterize all such peaks and divots of $L$.

A note on the alternating sums of powers of consecutive q-integers

In this paper we construct a new q-Euler numbers and polynomials. By using these numbers and polynomials, we give the interesting formulae related to alternating sums of powers of consecutive q-integers following an idea due to Euler.

An exponential Diophantine equation related to the difference between powers of two consecutive Balancing numbers

In this paper, we find all solutions of the exponential Diophantine equation $B_{n+1}^x-B_n^x=B_m$ in positive integer variables $(m, n, x)$, where $B_k$ is the $k$-th term of the Balancing sequence.

On consecutive 1's in continued fractions expansions of square roots of prime numbers

In this note, we study the problem of existence of sequences of consecutive 1's in the periodic part of the continued fractions expansions of square roots of primes. We prove unconditionally that, for a given $N\gg 1$, there are at least $N\log^{-3/2}N$ prime numbers $p\leq N$ such that the continued fraction expansion of $\sqrt{p}$ contains three consecutive 1's on the beginning of the periodic part. We also present results of our computations related to the considered problem and some related problems, formulate several open questions and conjectures and get some results under the assumption…

Sums of Consecutive Prime Squares

We prove explicit bounds for the number of sums of consecutive prime squares below a given magnitude.

Consecutive square-free values of the form $\mathbf{[αp], [αp]+1}$

In this short paper we shall prove that there exist infinitely many consecutive square-free numbers of the form $[αp]$, $[αp]+1$, where $p$ is prime and $α>0$ is irrational algebraic number. We also establish an asymptotic formula for the number of such square-free pairs when $p$ does not exceed given sufficiently large positive integer.

The Arithmetic of Consecutive Polynomial Sequences over Finite Fields

Motivated by a question of van der Poorten about the existence of infinite chain of prime numbers (with respect to some base), in this paper we advance the study of sequences of consecutive polynomials whose coefficients are chosen consecutively from a sequence in a finite field of odd prime characteristic. We study the arithmetic of such sequences, including bounds for the largest degree of irreducible factors, the number of irreducible factors, as well as for the number of such sequences of fixed length in which all the polynomials are irreducible.

On consecutive abundant numbers

A positive integer $n$ is called an abundant number if $σ(n)\ge 2n$, where $σ(n)$ is the sum of all positive divisors of $n$. Let $E(x)$ be the largest number of consecutive abundant numbers not exceeding $x$. In 1935, P. Erd\H os proved that there are two positive constants $c_1$ and $c_2$ such that $c_1\log\log\log x\le E(x)\le c_2\log\log\log x$. In this paper, we resolve this old problem by proving that, $E(x)/\log \log\log x$ tends to a limit as $x\to +\infty$, and the limit value has an explicit form which is between $3$ and $4$.

最简三次域与连续双二次域中的大Pólya群

Large Pólya groups in simplest cubic fields and consecutive bi-qaudratic fields

本文将数域的连续性概念推广到循环三次域和双二次域的参数族，证明存在无穷多个具有大Pólya群的连续域，推广了全实双二次域的已有结果。

Repdigits作为连续移位Tribonacci数的乘积

Repdigits as Product of Consecutive Shifted Tribonacci Numbers

研究连续移位Tribonacci数乘积中的repdigits，通过求解指数丢番图方程，完全刻画了所有解。

Three consecutive near-square squarefree numbers

In this note, we prove by using T. Estermann's and S. Dimitrov's arguments with an elementary inequality that there are infinitely many $n$ for which all of the numbers $n^2+1,n^2+2$ and $n^2+3$ are squarefree. We also improve the error term slightly in the case of two consecutive numbers of the same form, so that we are able to prove the following asymptotic formula. \begin{align*} \sum_{n\le X}μ^2(n^2+1)μ^2(n^2+2)μ^2(n^2+3)\sim\dfrac{7}{18}\prod_{p>3}\left(1-\dfrac{3+\left(\frac{-1}{p}\right)+\left(\frac{-2}{p}\right)+\left(\frac{-3}{p}\right)}{p^2}\right)X. \end{align*}

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

具有相同约数个数的连续整数段

Runs of Consecutive Integers Having the Same Number of Divisors

本文研究小于 N 且约数个数相同的连续整数最长段的长度上界，先用初等方法得到指数 λ=1/2，再利用 Jacobsthal 函数估计改进到 λ=1/3。

Large gaps between consecutive prime numbers

Let $G(X)$ denote the size of the largest gap between consecutive primes below $X$. Answering a question of Erdos, we show that $$G(X) \geq f(X) \frac{\log X \log \log X \log \log \log \log X}{(\log \log \log X)^2},$$ where $f(X)$ is a function tending to infinity with $X$. Our proof combines existing arguments with a random construction covering a set of primes by arithmetic progressions. As such, we rely on recent work on the existence and distribution of long arithmetic progressions consisting entirely of primes.

关于三个连续幂数的初等注记

An elementary note on three consecutive powerful numbers

本文利用初等方法及相关数论结果，推广了T. H. Chan关于三个连续幂数的结论，讨论了若干丢番图方程的解。

关于Erdős和Mirsky关于连续整数的因数个数之比的问题

On a Problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers

针对Erdős和Mirsky关于连续整数因数个数之比的问题，研究了满足存在无穷多整数$n$使$\alpha=\log\frac{d(n+1)}{d(n)}$的实数$\alpha$集合的闭包$\mathcal{L}$，并给出其密度的改进下界。

具有大类数的连续实二次域

Consecutive real quadratic fields with large class numbers

本文证明对任意正整数 $k$，存在大量连续整数 $d$，使 $k$ 个相邻实二次域的类数均达到本质上的最大可能量级。

Some heuristics on the gaps between consecutive primes

We propose the formula for the number of pairs of consecutive primes $p_n, p_{n+1}<x$ separated by gap $d=p_{n+1}-p_n$ expressed directly by the number of all primes $<x$, i.e. by $π(x)$. As the application of this formula we formulate 7 conjectures, among others for the maximal gap between two consecutive primes smaller than $x$, for the generalized Brun's constants and the first occurrence of a given gap $d$. Also the leading term $\log \log(x)$ in the prime harmonic sum is reproduced from our guesses correctly. These conjectures are supported by the computer data.

Pairs of k-free Numbers, consecutive square-full Numbers

We consider the error term of the asymptotic formula for the number of pairs of $k$-free integers up to $x$. Our error term improves results by Heath-Brown, Brandes and Dietmann/Marmon. We then extend our results to $r$-tuples of $k$-free numbers and improve previous results by Tsang. Furthermore, we establish an error term for consecutive square-full integers. Finally, we will show that for all $θ<3$ and for almost all $D$, the fundamental solution $ε_D$ associated to the Pell equation $x^2-Dy^2=1$ satisfies $ε_D> D^θ$. This improves/recovers previous results by Fouvry and Jouve. The main too…

A note on the sums of powers of consecutive q-integers

In this paper we construct the q-analogue of Barnes' Bernoulli numbers and plynomials of degree 2, which is an answer to a part of Schlosser's question. Finally, we treat the q-analogue of the sums of powers of consecutive integrs.

On Zeckendorf Related Partitions Using the Lucas Sequence

Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such partitions need not be unique. In this paper, we prove that a natural number can have at most two distinct non-consecutive partitions in the Lucas sequence, find all positive integers with a fixed term in their partition, and calculate the limiting value of the proportion of natural numbers that are not uniquely partitioned into the sum of non-consecutive te…

Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=μM+1$ with $μ=\left(η/δ\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,η,δ$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $η\equiv1\left(mod\,2\right)$ and $δ\equiv0,1$ or $5\left(mod\,6\right)$, and if $δ\equiv0\left(m…

算术多边形与连续平方和

Arithmetic Polygons and Sums of Consecutive Squares

本文研究了算术多边形与连续平方和的关系，证明了对于奇数N≥3存在至少一个N边算术多边形，并展示了存在无限多个偶数边的算术多边形。

Diophantine Inequalities with Primes, Auxiliary Inequalities, Evaluations of the Difference between Consecutive Primes

The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre - Ingham's problem on primes in intervals $((n - 1)^k, n^k)$ and also a problem on primes in intervals $(\frac{k-1}{k}n, \frac{k}{k-1}n)$ when $k$ is a real number. A number of the new results including an alternative proof of Ingham's theorem with the effectively computable constant and also Ingham's theorem with two primes are proved.

Note on the sums of powers of consecutive $q$-integers

In this paper we construct the $q$-analogue of Barnes's Bernoulli numbers and polynomials of degree 2, for positive even integers, which is an answer to a part of Schlosser's question. For positive odd integers, Schlosser's question is still open. Finally, we will treat the $q$-analogue of the sums of powers of consecutive integers.

Gaps of powers of consecutive primes and some consequences

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many values of $n$. As it is shown, the key ingredient to obtain this behaviour is a consequence of an extension of the Kummer's characterization of convergent series of positive terms.

关于连续无平方因子数之间间隙的矩

On moments of gaps between consecutive squarefree numbers

本文研究了连续无平方因子数之间间隙的矩，证明了相关渐近式在 $0 \le γ< 3.75$ 范围内成立，将此前 $0 \le γ< 3.6875$ 的最佳范围予以改进。

On the $p$-rationality of consecutive quadratic fields

In 2016, in the work related to Galois representations, Greenberg conjectured the existence of multi-quadratic $p$-rational number fields of degree $2^{t}$ for any odd prime number $p$ and any integer $t \geq 1$. Using the criteria provided by him to check $p$-rationality for abelian number fields, certain infinite families of quadratic, biquadratic and triquadratic $p$-rational fields have been shown to exist in recent years. In this article, for any integer $k \geq 1$, we build upon the existing work and prove the existence of infinitely many prime numbers $p$ for which the imaginary quadrat…

斐波那契数作为 $k$-广义斐波那契数列中连续项之和

Fibonacci Numbers as Sums of Consecutive Terms in $k$-Generalized Fibonacci Sequence

本文研究 $k$-广义斐波那契数列连续项之和何时为斐波那契数，证明对 $k\geq3$ 此类情形仅有有限多个，并推出两类数列交集有限。

Consecutive primes and Beatty sequences

Fix irrational numbers $α,\hatα>1$ of finite type and real numbers $β,\hatβ\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hatαm+\hatβ\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(α\hatα)^{-1}π(x)+O\big(x(\log x)^{-3/2+ε}\big), $$ where $π(x)$ is the prime cou…

连乘积具有不寻常解剖结构的连续整数

Products of consecutive integers with unusual anatomy

本文研究了连续自然数区间中连乘积的性质，定义了bad、very bad和type F3区间，并给出了关于这些区间数量的渐进行为及相关方程的解的近似数量。