L. H. Gallardo, O. Rahavandrainy
We give necessary conditions satisfied by the set of odd prime divisors of binary perfect polynomials. This allows us to get a new characterization of all the known perfect binary polynomials.
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In this paper, we consider certain finite sums related to the "largest odd divisor", and we obtain, using simple ideas and recurrence relations, sharp upper and lower bounds for these sums.
For a positive integer $n$, let $σ(n)$ denote the sum of the positive divisors of $n$. Let $d$ be a proper divisor of $n$. We call $n$ a deficient-perfect number if $σ(n)=2n-d$. In this paper, we show that the only odd deficient-perfect number with four distinct prime divisors is $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.
In a recent paper, Merca posed three conjectures on congruences for specific convolutions of a sum of odd divisor functions with a generating function for generalized $m$-gonal numbers. Extending Merca's work, we complete the proof of these conjectures.
In this paper, we consider a conjecture of Erdős and Rosenfeld when the number is a perfect square. In particular, we show that every perfect square $n$ can have at most five divisors between $\sqrt{n} - c \sqrt[4]{n}$ and $\sqrt{n} + c \sqrt[4]{n}$.
All bi-unitary perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors
该研究针对二元素域$\boldsymbol{\text{F}}_2$，找到了仅以梅森多项式为奇不可约因子的所有非分裂双酉完美多项式，是经典算术问题的多项式变体研究成果。
The sum of the reciprocals of the prime divisors of an odd perfect or odd primitive non-deficient number
本文在奇完全数及更弱的奇本原非亏数假设下，证明了素因子倒数之和 T(n) 与乘积 H(n) 关于最小素因子的新上界。
We investigate the implications of a curious biconditional involving divisors of odd perfect numbers, if Dris conjecture that $q^k < n$ holds, where $q^k n^2$ is an odd perfect number with Euler prime $q$. We then show that this biconditional holds unconditionally. Lastly, we prove that the inequality $q<n$ holds unconditionally.
The only (unitary) perfect polynomials over $\mathbb{F}_2$ that are products of $x$, $x+1$ and Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$. Other consequences of such a factorization are new results about odd perfect polynomials.
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 …
If $N = {q^k}{n^2}$ is an odd perfect number, where $q$ is the Euler prime, then we show that $n < q$ is sufficient for Sorli's conjecture that $k = ν_{q}(N) = 1$ to hold. We also prove that $q^k < 2/3{n^2}$, and that $I(q^k) < I(n)$, where $I(x)$ is the abundancy index of $x$.
We address an arithmetic problem in the ring $\F_2[x]$ related to the fixed points of the sum of divisors function. We study some binary polynomials $A$ such that $σ(A)/A $ is still a binary polynomial. Technically, we prove that the only (unitary) perfect polynomials over $\F_2$ that are products of $x$, $x+1$ and of Mersenne primes are precisely the nine (resp. nine "classes") known ones. This follows from a new result about the factorization of $M^{2h+1} +1$, for a Mersenne prime $M$ and for a positive integer $h$.
The divisor function $σ(n)$ sums the divisors of $n$. We call $n$ abundant when $σ(n) - n > n$ and perfect when $σ(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor function motivates their recursive analogs, which I introduce here. A number is recursively abundant, or ample, if $a(n) > n$ and recu…
We call positive integer n a near-perfect number, if it is sum of all its proper divisors, except of one of them ("redundant divisor"). We prove an Euclid-like theorem for near-perfect numbers and obtain some other results for them.
Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfec…
Let $n$ and $k$ be positive integers and $σ(n)$ the sum of all positive divisors of $n$. We call $n$ an exactly $k$-deficient-perfect number with deficient divisors $d_1, d_2, \ldots, d_k$ if $d_1, d_2, \ldots, d_k$ are distinct proper divisors of $n$ and $σ(n)=2n-(d_1+d_2+\ldots + d_k)$. In this article, we show that the only odd exactly $3$-deficient-perfect number with at most two distinct prime factors is $1521=3^2 \cdot 13^2$.
On the Asymptotic Behavior of a Multiplicative Arithmetic Function Related to the Divisor Function Over Perfect Squares Integers Generated by Shifting
本文研究了与完美平方整数相关的除数函数的渐进行为，推导了特定求和式的渐近公式。
Non-splitting bi-unitary perfect polynomials over $\mathbb{F}_4$ with less than five prime factors
本文确定了有限域 $\mathbb{F}_4$ 上所有至多包含四个不可约因子的非分裂双酉完美多项式，并指出此类因子有无限多个。
For a positive integer $n$, if $σ(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $σ(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.
For a radical extension K of odd prime degree the ring O_K of integers is constructed as a product of subrings with the following property: for all prime divisors q of the discriminant of O_K there is a q-maximal factor. The discriminant of O_K is the greatest common divisor of the discriminants of all factors. The results are applied to give a criterion for the monogeneity of K where the opposite is not true.
We call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $σ(k)(m+1)=2n$, where $σ$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect numbers could be established for spoof odd perfect numbers (otherwise known in the literature as Descartes numbers). In particular, we prove that $k$ is not almost perfect.
A Divisor-Sum Analogue of the Collatz Map
本文提出一个除数和函数的 Collatz 映射类比，猜想所有轨道到达 1，并证明单次应用该映射对几乎所有奇数有对数下降估计，但估计不可迭代。
A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides an upper bound for density of odd integers which could satisfy ${{σ(N)}\over N}=2$, where $N$ belongs to a fixed interval with a lower limit greater than $10^{300}$. Characteristics of prime divisors of repunits are used to establish whether the product containing the repunits can be a perfect square. It is shown that the arithmetic primitive divisors with …
The divisor function $σ(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for $ν_{p}(σ(n))$ are established. For $p=2$, these involve only the odd primes dividing $n$. These expressions are used to establish the bound $ν_{2}(σ(n)) \leq \lceil\log_{2}(n) \rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is $ν_{p}(σ(n)) \leq \lceil \log_{p}(n) \rceil$, with equality related to solutions…
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 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 extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.
We show that if {1, b, c, d} is a D(-1) diophantine quadruple with b<c<d and c=1+s^2, then the cases s=p^k, s=2p^k, c=p and c=2p^k do not occur, where p is an odd prime and k is a positive integer. For the integer d=1+x^2, we show that it is not prime and that x is divisible by at least two distinct odd primes. Furthermore, we present several infinite families of integers b such that the D(-1) pair {1, b} cannot be extended to a D(-1) quadruple. For instance, we show that if r=5p where p is an odd prime, then the D(-1) pair {1, r^2+1} cannot be extended to a D(-1) quadruple.
Let $Ω(n)$ denote the total number of prime divisors of $n$ (counting multiplicity) and let $ω(n)$ denote the number of distinct prime divisors of $n$. Various inequalities have been proved relating $ω(N)$ and $Ω(N)$ when $N$ is an odd perfect number. We improve on these inequalities. In particular, we show that if $3 \not| N$, then $Ω\geq \frac{8}{3}ω(N)-\frac{7}{3}$ and if $3 |N$ then $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$
Let $k\ge2$ be an integer. A natural number $n$ is called $k$-perfect if $σ(n)=kn.$ For any integer $r\ge1$ we prove that the number of odd $k$-perfect numbers with at most $r$ distinct prime factors is bounded by $k4^{r^3}$.
We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, ``spoof'' perfect factorization $3^2\cdot 7^2\cdot 11^2\cdot 13^2\cdot 22021^1$. More recently, Voight found the spoof perfect factorization $3^4\cdot 7^2\cdot 11^2\cdot 19^2\cdot(-127)^1$. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases -- there are twenty-one in total. We show that the structure of odd, spoof perfect factorizations …
We analyze a natural parity problem on the divisors associated to Sturmian words. We find a clear bias towards the odd divisors and obtain a sharp asymptotic estimate for the average of the difference odd-even function tamed by a mollifier, which proves various experimental results.
Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the sums of reciprocals of the factors of odd perfect numbers, or, in more technical terms, the properties of the sub-sums of σ_{-1} (n). By this result, it also establishes strong bounds on the prime factors of odd perfect numbers using the number of distinct prime factors it may possess.
The existence of a perfect odd number is an old open problem of number theory. An Euler's theorem states that if an odd integer $ n $ is perfect, then $ n $ is written as $ n = p ^ rm ^ 2 $, where $ r, m $ are odd numbers, $ p $ is a prime number of the form $ 4 k + 1 $ and $ (p, m) = 1 $, where $ (x, y) $ denotes the greatest common divisor of $ x $ and $ y $. In this article we show that the exponent $ r $, of $ p $, in this equation, is necessarily equal to 1. That is, if $ n $ is an odd perfect number, then $ n $ is written as $ n = pm ^ 2. $
We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{σ(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that if $D=pq$ denotes a spoof odd perfect number other than Descartes' example, with pseudo-prime factor $p$, then $q>10^{12}$. Furthermore, we find irregularities in the ending digits of integers $n\in\mathcal{S}$ and study aspects of its density, leading us to conjecture that the amount of numbers in $\mathcal{S}$ below $k$ is $\sim10 \log(k)$.
On odd perfect numbers of special forms
本文研究了特殊乘法形式的奇数族成为完全数的必要条件，推广了Steuerwald、Kanold和McDaniel等人的早期工作。
We prove there exist infinitely many odd integers $n$ for which there exists a pair of positive divisors $d_1, d_2>1$ of $(n^2+1)/2$ such that $$d_1+d_2=δn+(δ+2).$$ We prove the similar result for $\varepsilon=δ-2$ and $δ\equiv4, 6\pmod{8}$ using different approaches and methods.
We study the average value of the divisor function $τ(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes $p$ and any fixed integer $k\ge 7$.
The study of perfect numbers (numbers which equal the sum of their proper divisors) goes back to antiquity, and is responsible for some of the oldest and most popular conjectures in number theory. We investigate a generalization introduced by Pollack and Shevelev: $k$-near-perfect numbers. These are examples to the well-known pseudoperfect numbers first defined by Sierpiński, and are numbers such that the sum of all but at most $k$ of its proper divisors equals the number. We establish their asymptotic order for all integers $k\ge 4$, as well as some properties of related quantities.
The Descartes-Frenicle-Sorli conjecture predicts that $k=1$ if $q^k n^2$ is an odd perfect number with Euler prime $q$. In this note, we present some conditions equivalent to this conjecture.
The Euler's form of odd perfect numbers, if any, is $n=π^αN^2$, where $π$ is prime, $(π,N)=1$ and $π\equiv α\equiv 1 \pmod{4}$. Dris conjecture states that $N>π^α$. We find that $N^2>\frac{1}{2}π^γ$, with $γ=max\{ω(n)-1,α\}$; $ω(n)\geq 9$ is the number of distinct prime factors of $n$.
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give a short proof for this conjecture.
Euler showed that if an odd perfect number exists, it must be of the form $N = p^αq_{1}^{2β_{1}}$ $\ldots$ $q_{k}^{2β_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $α$, $β_{i} \geq 1$, for $1 \leq i \leq k$, with $p \equiv α\equiv 1 \pmod{4}$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $β_{i} \equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ and show that $N$ is not perfect, simply, if each $β_{i} \equiv 2 \pmod{5}$.
Let $N$ be an odd perfect number. Let $ω(N)$ be the number of distinct prime factors of $N$ and let $Ω(N)$ be the total number (counting multiplicity) of prime factors of $N$. We prove that $\frac{99}{37}ω(N) - \frac{187}{37} \leq Ω(N)$ and that if $3\nmid N$, then $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.
We shall show that, for any given primes $\ell\geq 17$ and $p, q\equiv 1\pmod{\ell}$, the diophantine equation $(x^\ell-1)/(x-1)=p^m q$ has at most four positive integral solutions $(x, m)$ and give its application to odd perfect number problem.
We shall given a new effectively computable upper bound of odd perfect numbers whose Euler factors are powers of fixed exponent, improving our old result in T. Yamada, Colloq. Math. 103 (2005), 303--307.
A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $σ(n)=kn$, where $σ(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=π^αM^2$ such that $π\equivα(\text{mod}8)$. We also present some examples to show the nonexistence of odd perfect numbers as applications.
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
In this note, we show that if $N$ is an odd perfect number and $q^α$ is some prime power exactly dividing it, then $σ(N/q^α)/q^α>5$. In general, we also show that if $σ(N/q^α)/q^α<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.
A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.
If \(A \) is a set of natural numbers containing \(0 \), then there is a unique nonempty "reciprocal" set \(B \) of natural numbers (containing \(0 \)) such that every positive integer can be written in the form \(a + b \), where \(a \in A \) and \(b \in B \), in an even number of ways. Furthermore, the generating functions for \(A \) and \(B \) over \(\FF_2 \) are reciprocals in \(\FF_2 [[q]] \). We consider the reciprocal set \(B \) for the set \(A \) containing \(0 \) and all integers such that \(σ(n) \) is odd, where \(σ(n) \) is the sum of all the positive divisors of \(n \). This problem…
We study equations of the form $σ(p^{q-1})=Az$, where $p$ is a prime, $q$ is a fixed odd prime, $A$ is a fixed integer and $z$ is an integer composed of primes in a fixed finite set. We shall improve upper bounds for the size and the number of solutions of such equations.
If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then the Descartes-Frenicle-Sorli conjecture predicts that $k=1$. Brown has recently announced a proof for the inequality $q < n$, and a partial proof that $q^k < n$ holds under many cases. In this article, we give a strategy for strengthening Brown's result to $q^2 < n$.
Let $p^k m^2$ be an odd perfect number with special prime $p$. In this article, we provide an alternative proof for the biconditional that $σ(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $σ(m^2)/p^k$ is a square.
If $N = {p^k}{m^2}$ is an odd perfect number with special prime factor $p$, then it is proved that ${p^k} < (2/3){m^2}$. Numerical results on the abundancy indices $\frac{σ(p^k)}{p^k}$ and $\frac{σ(m^2)}{m^2}$, and the ratios $\frac{σ(p^k)}{m^2}$ and $\frac{σ(m^2)}{p^k}$, are used. It is also showed that $m^2 > \frac{\sqrt{6}}{2}({10}^{150})$.
Let $σ(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $σ(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^α\prod_{j=1}^k q_j^{2 β_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv α\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $Ω(N) := α+ 2 \sum_{j=1}^k β_j$. Sayers showed that $Ω(N) \geq 29$. This was later extended by Iannucci and Sorli to show that $Ω(N) \geq 37$. This was extended by the author to show that $Ω(N) \geq 47$…
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.
We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{α_1}... p_s^{α_s} q_1^{2β_1}... q_t^{2β_t}$ with $β_1, ..., β_t$ in some finite set S satisfies $σ(N)=\frac{n}{d}N$, then N has a prime factor smaller than C, where C is an effective computable constant depending only on s, n, S.
We examine the existence of rational divisors on modular curves of $\mathcal{D}$-elliptic sheaves and on Atkin-Lehner quotients of these curves over local fields. Using a criterion of Poonen and Stoll, we show that in infinitely many cases the Tate-Shafarevich groups of the Jacobians of these Atkin-Lehner quotients have non-square orders.
Let $p^k m^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m < p^k$ follows from $m^2 - p^k = 2^r t$, where $r \geq 2$ and $\gcd(2,t)=1$, under the following hypotheses: (a) $m > t > 2^r$, or (b) $m > 2^r > t$. We also prove that the estimate $m^2 - p^k > 2m$ holds. We can also improve this unconditional estimate to $m^2 - p^k > {313m^2}/315$.
The paper is about an arithmetic problem in $\F_2[x]$. We give \emph{admissible} (necessary) conditions satisfied by a set of odd prime divisors of perfect polynomials over $\F_2$. This allows us to prove a new characterization of \emph{all} known perfect polynomials, and to open a way of finding more of them (if they exist).
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