We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bretèche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Deshouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
This paper studies the largest cycles consisted by the quadratic residues modulo prime numbers. We give some formulae about the maximum length of the cycles. Especially, the formula for modulo Fermat primes is given.
In 1982, Erdös and van Lint showed an estimate for the average of the ratio of the smallest and largest prime divisor of $n$. In this note, we apply C.H. Jia's method to give an estimate for the average of positive integer power of the ratio.
We consider the distribution of the largest prime divisor of the integers in the interval $[2,x]$, and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as $x$ tends to infinity, we look at the set of those prime numbers which, for some value of $x$, occur most frequently as the largest prime divisor of the integers in the interval $[2,x]$. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some poin…
Improving on a theorem of Heath-Brown, we show that if $X$ is sufficiently large then a positive proportion of the values $\{n^3+2:n\in (X,2X]\}$ have a prime factor larger than $X^{1+10^{-52}}$.
In this paper, we find all integers $x$ such that $x^{2}-1$ has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer $n$ we can find the largest positive integer $x$ such that all prime factors of each of $x, x+1,..., x+n$ are less than 100.
Today, prime numbers attained exceptional situation in the area of numbers theory and cryptography. As we know, the trend for accessing to the largest prime numbers due to using Mersenne theorem, although resulted in vast development of related numbers, however it has reduced the speed of accessing to prime numbers from one to five years. This paper could attain to theorems that are more extended than Mersenne theorem with accelerating the speed of accessing to prime numbers. Since that time, the reason for frequently using Mersenne theorem was that no one could find an efficient formula for a…
Let P(k) be the largest prime factor of the positive integer k. In this paper, we prove that the series $\sum_{n\ge 1}\frac{(\log n)^a}{P(2^n-1)}$ is convergent for each constant a<1/2, which gives a more precise form of a result of C. L. Stewart from 1977.
Let $P(X)\in\mathbb{Z}[X]$ be an irreducible, monic, quartic polynomial with cyclic or dihedral Galois group. We prove that there exists a constant $c_P>0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor $\ge n^{1+c_P}$.
We consider the Dickman function $Ψ(x,y)$ in the limit when $\ln x/\ln y\to \infty $ and $\ln \ln x \ln y\to 0$. The asymptotic value is expressed in terms of the ratio of iterated loragithm of $x$ and $ln y$.
It was conjectured by Gauss that any negative discriminant with at least 32 genera has at least two classes of binary quadratic forms in each genus. We prove that such a discriminant cannot have a particularly large prime factor, by an argument similar to that of Baker's solution to Gauss' class number one problem.
Let $N$ be an odd perfect number and let $a$ be its third largest prime divisor, $b$ be the second largest prime divisor, and $c$ be its largest prime divisor. We discuss steps towards obtaining a non-trivial upper bound on $a$, as well as the closely related problem of improving bounds $bc$, and $abc$. In particular, we prove two results. First we prove a new general bound on any prime divisor of an odd perfect number and obtain as a corollary of that bound that $$a < 2N^{\frac{1}{6}}.$$ Second, we show that $$abc < (2N)^{\frac{3}{5}}.$$ We also show how in certain circumstances these bounds …
Let $P(m)$ denote the largest prime factor of an integer $m\geq 2$, and put $P(0)=P(1)=1$. For an integer $k\geq 2$, let $(F_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Fibonacci sequence which starts with $0,...,0,1$ ($k$ terms) and each term afterwards is the sum of the $k$ preceding terms. Here, we show that if $n\geq k+2$, then $P(F_n^{(k)})>c\log\log n$, where $c>0$ is an effectively computable constant. Furthermore, we determine all the $k-$Fibonacci numbers $F_n^{(k)}$ whose largest prime factor is less than or equal to 7.
For an integer $m >1$, we denote by $P(m)$ the largest prime divisor of $m$. We prove that $\limsup_{n \rightarrow +\infty} P(n!+1)/n \geqslant 1+9\log 2>7.238$, which improves a result of Stewart. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup_{n \rightarrow +\infty} P(n!+f(n))/n \geqslant 1+9\log2$. This improves a result of Luca and Shparlinski. These improvements come from an additional combinatoric idea to the works mentioned above.
Let $\{p_j(n)\}_{j=1}^{ω(n)}$ denote the increasing sequence of distinct prime factors of an integer $n$. We provide details for the proof of a statement of Erdős implying that, for any function $ξ(n)$ tending to infinity with $n$, we have $$f(n):=\max_{1\leqslant j<ω(n)}\log \Big({\log p_{j+1}(n)\over \log p_j(n)}\Big)=\log_3n+O(ξ(n))$$ for almost all integers $n$.
Let $M(x)$ denote the median largest prime factor of the integers in the interval $[1,x]$. We prove that $$M(x)=x^{\frac{1}{\sqrt{e}}\exp(-\text{li}_{f}(x)/x)}+O_ε(x^{\frac{1}{\sqrt{e}}}e^{-c(\log x)^{3/5-ε}})$$ where $\text{li}_{f}(x)=\int_{2}^{x}\frac{\{x/t\}}{\log t}dt$. From this, we obtain the asymptotic $$M(x)=e^{\frac{γ-1}{\sqrt{e}}}x^{\frac{1}{\sqrt{e}}}(1+O(\frac{1}{\log x})),$$ where $γ$ is the Euler Mascheroni constant. This answers a question posed by Martin, and improves a result of Selfridge and Wunderlich.
We define $g(n)$ to be the maximal order of an element of the symmetric group on $n$ elements. Results about the prime factorization of $g(n)$ allow a reduction of the upper bound on the largest prime divisor of $g(n)$ to $1.328\sqrt{n\log n}$.
We show that there are infinitely many primes $p$ such that $p-1$ is divisible by a square $d^2 \geq p^θ$ for $θ=1/2+1/2000.$ This improves the work of Matomäki (2009) who obtained the result for $θ=1/2-\varepsilon$ (with the added constraint that $d$ is also a prime), which improved the result of Baier and Zhao (2006) with $θ=4/9-\varepsilon.$ Similarly as in the work of Matomäki, we apply Harman's sieve method to detect primes $p \equiv 1 \, (d^2)$. To break the $θ=1/2$ barrier we prove a new bilinear equidistribution estimate modulo smooth square moduli $d^2$ by using a similar argument as …
Acquaah and Konyagin showed that if $N$ is an odd perfect number where $N= p_1^{a_1}p_2^{a_2} \cdots p_k^{a_k}$ where $p_1 < p_2 \cdots < p_k$ then one must have $p_k < 3^{1/3}N^{1/3}$. Using methods similar to theirs, we show that $p_{k-1}< (2N)^{1/5}$ and that $p_{k-1}p_k < 6^{1/4}N^{1/2}.$ We also show that if $p_k$ and $p_{k-1}$ are close to each other than these bounds can be further strengthened.
For $k\ge 2$ even, let $d_{k,N}$ denote the dimension of the largest simple Hecke submodule of $S_{k}(Γ_0(N); \mathbb{Q})^\text{new}$. We show, using a simple analytic method, that $d_{k,N} \gg_k \log\log N / \log(2p)$ with $p$ the smallest prime co-prime to $N$. Previously, bounds of this quality were only known for $N$ in certain subsets of the primes. We also establish similar (and sometimes stronger) results concerning $S_{k}(Γ_0(N), χ)$, with $k \geq 2$ an integer and $χ$ an arbitrary nebentypus.
The notion of chiral prime concatenations is studied as a recursive construction of prime numbers starting from a seed set and with appropriate blocks to define the primality growth, generation by generation, either from the right or from the left. Several basic questions are addressed like whether chiral concatenation is a symmetric process, an endless process, as well as the calculation of largest chiral prime numbers. In particular, the largest left-concatenated prime number is constructed. It is a unique prime number of 24 digits. By introducing anomalous left-concatenations of primes we c…
Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let $\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost surely have $$\bigg|\sum_{\substack{n\leq x\\ P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where $P(n)$ stands for the largest prime factor of $n$. This gives an indication of the almost sure size of the largest fluctuations of $f$.
The paper is concerned with estimating the number of integers smaller than $x$ whose largest prime divisor is smaller than $y$, denoted $ψ(x,y)$. Much of the related literature is concerned with approximating $ψ(x,y)$ by Dickman's function $ρ(u)$, where $u=\ln x/\ln y$. A typical such result is that $$ ψ(x,y)=xρ(u)(1+o(1)) \eqno (1) $$ in a certain domain of the parameters $x$ and $y$. In this paper a different type of approximation of $ψ(x,y)$, using iterated logarithms of $x$ and $y$, is presented. We establish that $$ \ln (\frac ψ{x})=-u [\ln ^{(2)}x-\ln ^{(2)}y+\ln ^{(3)}x-\ln ^{(3)}y+\ln …
We show that for each prime p > 7, every residue mod p can be represented by a squarefree number with largest prime factor at most p. We give two applications to recursive prime generators akin to the one Euclid used to prove the infinitude of primes.
Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim3p_n$. Similarly, for the largest integer $f_n$ not contained in $S_n$, by computational evidence we suspect that $\bullet$ $f_n$ is an odd number for $n\geq5$ and $\bullet$ $f_n\sim3p_n$; further $\bullet$ $4p_n>f_{n+1}$ for $n\geq1$. If $f_n$ is odd for large $n$, then $f_n\sim3p_n$. In case $f_n\sim3p_n$ every large even integer $x$ is the sum of two primes. If …
Let $\mathcal{P}$ be the set of all primes and $π(x)$ be the number of primes up to $x$. For any $n\ge 2$, let $P^+(n)$ be the largest prime factor of $n$. For $0<c<1$, let $$T_c(x)=\#\{p\le x:p\in \mathcal{P},P^+(p-1)\ge p^c\}.$$ In this note, we proved that there exists some $c<1$ such that $$\limsup_{x\rightarrow\infty}\frac{T_c(x)}{π(x)}<\frac12,$$ which disproves a conjecture of Chen and Chen.
It is proven that there are infinitely prime pairs whose difference is no greater than 84. In proving that result, a new sieve weight is constructed. The new idea is to construct the largest sieve support which admits the use of the Bombieri-Vinogradov theorem.
We estimate from below the lower density of the set of prime numbers p such that p-1 has a prime factor of size at least p^c, where c lies in between 1/4 and 1/2. We also establish upper and lower bounds on the counting function of the set of positive integers n up to x with exactly k prime factors, counted with or without multiplicity, such that the largest prime factor of gcd(p-1 : p | n) exceeds n^{1/2k}.
The degree of insulation of a prime $p$ is defined as the largest interval around it in which no other prime exists. Based on this, the $n$-th prime $p_{n}$ is said to be insulated if and only if its degree of insulation is higher than its neighboring primes. Consequently, a new special sequence emerges given as 7, 13, 23, 37, 53, 67, 89, 103, 113, 131, 139, 157, 173, 181, 193, 211, 233, 277, 293, and so on. This paper presents several properties and intriguing relations concerning degree of insulation and insulated primes. Finally, the reader is left with a captivating open problem.
We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erdös, Odlyzko and Sárközy.
The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid ϕ(n)$, where $ϕ$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log n/\log\log\log n$. That is, pr$(n) \sim \log\log n/\log\log\log n$ as $n\to\infty$ on a set of integers of asymptotic density 1. In fact we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer $k$ with $k!\mid ϕ(n)$.
We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.
We prove some new lower bounds for the counting function $\mathcal N_{\mathcal C}(x)$ of the set of Novák-Carmichael numbers. Our estimates depend on the bounds for the number of shifted primes without large prime factors. In particular, we prove that $\mathcal N_{\mathcal C}(x) \gg x^{0.7039-o(1)}$ unconditionally and that $\mathcal N_{\mathcal C}(x) \gg xe^{-(7+o(1))(\log x)\frac{\log\log\log x}{\log\log x}}$, under some reasonable hypothesis.
An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $Ω$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $Ω=6$, count all PANs and square-free PANs up to $Ω=7$ and count all odd PANs and odd square-free PANs up to $Ω=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example exist…
