On the Existence of Odd Perfect Numbers

Odd, spoof perfect factorizations

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 …

Bounds On Factors Of Odd Perfect Numbers

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.

Where do odd perfect numbers live?

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. $

On the Density of Spoof Odd Perfect Numbers

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 $\sim10 \log(k)$.

关于特殊形式的奇完全数

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。

On odd deficient-perfect numbers with four distinct prime divisors

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 $3^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}$.

On (unitary) perfect polynomials over $\mathbb{F}_2$ with only Mersenne primes as odd divisors

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.

Conditions Equivalent to the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers

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.

On Dris Conjecture about Odd Perfect Numbers

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 $n$.

A short proof for the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

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.

More on the Nonexistence of Odd Perfect Numbers of a Certain Form

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 $β_{i} \equiv 2 \pmod{5}$.

On inequalities involving counts of the prime factors of an odd perfect number

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 $\frac{51}{19}ω(N)-\frac{46}{19} \leq Ω(N)$.

An exponential diophantine equation related to odd perfect numbers

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.

A new upper bound for odd perfect numbers of a special form

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.

Odd Multiperfect Numbers

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.

奇完全数或奇本原非亏数素因子倒数之和

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。

On a Curious Biconditional Involving Divisors of Odd Perfect Numbers

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.

A note on odd perfect numbers

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 $K$.

在仅以梅森素数为奇素因子的$\boldsymbol{\text{F}}_2$上的所有双酉完美多项式

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。

New Results for the Descartes-Frenicle-Sorli Conjecture on Odd Perfect Numbers

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 $q^2 < n$.

The Abundancy Index of Divisors of Odd Perfect Numbers

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 $x$.

Some Modular Considerations Regarding Odd Perfect Numbers

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.

An improvement of an inequality of Ochem and Rao concerning odd perfect numbers

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 $Ω(N) \geq \frac{21}{8}ω(N)-\frac{39}{8}.$

On the Components of an Odd Perfect Number

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 $m^2 > \frac{\sqrt{6}}{2}({10}^{150})$.

New techniques for bounds on the total number of Prime Factors of an Odd Perfect Number

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 $Ω(N) \geq 47$…

Upper bounds on the second largest prime factor of an odd perfect number

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.

On the divisibility of odd perfect numbers by a high power of a prime

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.

On the quantity $m^2-p^k$ where $p^k m^2$ is an odd perfect number -- Part II

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 $m^2 - p^k > {313m^2}/315$.

On the third largest prime divisor of an odd perfect number

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 …

New Results for Sorli's Conjecture on Odd Perfect Numbers - Part II

If $N={q^k}{n^2}$ is an odd perfect number given in Eulerian form, then Sorli's conjecture predicts that $k=ν_{q}(N)=1$. In this article, we give some further results related to this conjecture and those contained in the papers \cite{Dris} and \cite{Dris2}. (withdrawn because of a crucial gap in Theorem 1 [see https://arxiv.org/pdf/1309.0906.pdf for what is currently provable in this regard], as well as elementary mistakes in the numerical bounds from pages 2 to 4)

All even (unitary) perfect polynomials over $\F_2$ with only Mersenne primes as odd divisors

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 $h$.

On the number of total prime factors of an odd perfect number

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 of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}ω- \frac{286}{113} \leq Ω. $ If $3|N$, then $\frac{66}{25}ω-5\leqΩ.$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $ω(N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor.

On the Descartes-Frenicle-Sorli and Dris Conjectures Regarding Odd Perfect Numbers

Dris conjectured in his masters thesis that the inequality $q^k < n$ always holds, if $N = {q^k}{n^2}$ is an odd perfect number with special prime $q$. In this note, we initially show that either of the two conditions $n < q^k$ or $σ(q)/n < σ(n)/q$ holds. This is achieved by first proving that $σ(q)/n \neq σ(n)/q^k$, where $σ(x)$ is the sum of the divisors of $x$. Using this analysis, we further show that the condition $q < n < q^k$ holds in four out of a total of six cases. Finally, we prove that $n < q^k$, and that this holds unconditionally. This finding disproves both the Dris Conjecture a…

A Sufficient Condition for Disproving Descartes's Conjecture on Odd Perfect Numbers

Let $σ(x)$ be the sum of the divisors of $x$. If $N$ is odd and $σ(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. In this note, we show that $q < n$ implies that Descartes's conjecture (previously Sorli's conjecture), $k = ν_{q}(N) = 1$, is not true. This then implies an unconditional proof for the biconditional $$k = ν_{q}(N) = 1 \Longleftrightarrow n < q.$$ Lastly, following a recent result of Cohen and Sorli, we show that if $q < n$, then either $q > 5$ or $k > 5$ i…

A Rationality Condition for the Existence of Odd Perfect Numbers

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 …

A new approach to odd perfect numbers via GCDs

Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(σ(q^k),σ(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,σ(n^2)\bigg)$$ and $$I = \gcd\bigg(n,σ(n^2)\bigg).$$ We prove that $G \times H = I^2$. (Note that it is trivial to show that $G \mid I$ and $I \mid H$ both hold.) We then compute expressions for $G, H,$ and $I$ in terms of $σ(q^k)/2, n,$ and $\gcd\bigg(σ(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$, then $σ(q^k)/2$ is not squarefree. Other natural and related results are derived further. Lastly, we conjecture that the set $$\mathscr{A} = \{m : \g…

Some Extensions to Touchard's Theorem on Odd Perfect Numbers

The multiplicative structure of an odd perfect number $n$, if any, is $n=π^αM^2$, where $π$ is prime, $\gcd(π,M)=1$ and $π\equiv α\equiv1\pmod{4}$. An additive structure of $n$, established by Touchard, is that "$\bigl(n\equiv 9\pmod{36}\bigr )$ OR $\bigl (n\equiv1\pmod{12}\bigr )$". A first extension of Touchard's result is that the proposition "$\bigl(n\equiv x^2\pmod{4 x^2}\bigr )$ OR $\bigl (n\equiv π\equiv1\pmod{4 x}\bigr )$" holds for $x=3$ (the extension is due to the fact that the second congruence contains also $π$). We further extend the proof to $x=α+2$, $α+2$ prime, with the restri…

关于 Kanold 奇完全数定理的一个改进

On a theorem of Kanold on odd perfect numbers

本文改进 Kanold 关于奇完全数的经典定理，在特定素数分解和同余条件下，将整除因子从原有结论提升为 t^5 整除 N。

On the divisibility of odd perfect numbers, quasiperfect numbers and amicable numbers by a high power of a prime

We shall give an explicit upper bound for the smallest prime factor of multiperfect numbers of the form $N=p_1^{α_1}\cdots p_s^{α_s} q_1^{β_1}\cdots q_t^{β_t}$ with $β_1, \ldots, β_t$ bounded by a given constant. We shall also give similar results for quasiperfect numbers and relatively prime amicable pairs of opposite parity.

关于丢番图方程 $2ln^{2} = 1+q+ \cdots +q^α$ 的一个注记及其对奇完全数的应用

A note on the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^α$ and application to odd perfect numbers

本文利用二次序中的素理想分解和广义Fermat方程本原解，证明奇完全数中比值 $σ(n^{2})/q^α$ 在 $α\neq 1$ 时既非平方数也非平方数乘单素数，并推广到奇多重完全数。

Unitary super perfect numbers

We shall show that 9, 165 are all of the odd unitary super perfect numbers.

Recursively abundant and recursively perfect numbers

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…

Perfect forms and the Vandiver conjecture

Let p be an odd prime, n an odd positive integer and C the p-Sylow subgroup the class group of the p-cyclotomic extension of the rationals. When log(p) is bigger than n**(224n**4), we prove that the eigenspace on C attached to the (p-n)-th power of the Teichmuller character is trivial. The proof uses the K-theory of the integers and the Voronoi reduction theory of quadratic forms.

On Sinha's note on perfect numbers

We shall show that there is no odd perfect number of the form $2^n+1$ or $n^n+1$.

Some multiplicative functions over $\mathbb{F}_2$

We adapt (over $\mathbb{F}_2$) the general notions of multiplicative function, Dirichlet convolution and Inverse. We get some interesting results, namely necessary conditions for an odd binary polynomial to be perfect. Note that we are inspired by the "analogous" works in \cite{Gall-Rahav-newcongr} and \cite{Touchard}, about odd perfect numbers.

Solving the Odd Perfect Number Problem: Some Old and New Approaches

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $σ(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = σ(p^k)/p^k, Y = σ(m^2)/m^2$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.8…

A Partial Proof of a Conjecture of Dris

Euler showed that if an odd perfect number $N$ exists, it must consist of two parts $N=q^k n^2$, with $q$ prime, $q \equiv k \equiv 1 \pmod{4}$, and gcd$(q,n)=1$. Dris conjectured that $q^k < n$. We first show that $q<n$ for all odd perfect numbers. Afterwards, we show $q^k < n$ holds in many cases.

Perfect Powers of Five with Few Ternary Digits

In this note we will analyze a diophantine equation raised by Michael Bennett in [1] that is pivotal in establishing that powers of five has few digits in its ternary expansion. We will show that the Diophantine equation $3^{a}+3^{b}+2=n^5$, where $(n,3)=1$ and $a>b>0$ is insoluble for pairs of positive integers $(a,b)$ where they are both even or one is even and the other is odd. In the case where both $(a,b)$ are odd, there is one known solution $2^5=3^3+3^1+2.$ We will show that there are no other solutions to the diophantine equation for $n^{5}<32\left(1+3(10^6)\right)^5$.

On the existence of special elements in odd $K$-theory groups

Let $k$ be an imaginary quadratic number field, and $F/k$ a finite abelian extension of Galois group $G$. We investigate the relationship between the conjectural special elements introduced in \cite{Burns-DeJeu-Gangl} and ETNC in the semi-simple case. This provides a partial proof of the conjecture for $F/k$ under certain conditions.

On Deficient Perfect Numbers with Four Distinct Prime Factors

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.

奇伪多完全数

Odd Spoof Multiperfect Numbers

本研究将伪完全数定义推广到多完全数范畴，提出相应搜索算法，发现多个类似笛卡尔数的新奇伪多完全数（如8999757），并探讨了这类数的部分性质。

Solving the Odd Perfect Number Problem: Some New Approaches

A conjecture predicting an injective and surjective mapping $X = \displaystyle\frac{σ(p^k)}{p^k}, Y = \displaystyle\frac{σ(m^2)}{m^2}$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.85 < X + Y < 3$, is disproved. We will show that if an OPN $N$ has the form above, then $p^k < {2/3}{m^2}$. We then give a somewhat weaker corollary to this last result ($m^2 - p^k \ge 8$) and give possible improvements along these lines. We will also attempt to prove a conjectured improvement to $p^k < m$ by observ…

On Exactly $3$-Deficient-Perfect Numbers

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 $1521=3^2 \cdot 13^2$.

Euclid-Euler Heuristics for Perfect Numbers

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M = (2^p - 1)\cdot{2^{p - 1}}$ where $p$ and $2^p - 1$ are primes. In this article, we show how simple considerations surrounding the differences between the underlying properties of the Eulerian and Euclidean forms of perfect numbers give rise to what we will call the Euclid-Euler heuristics for perfect numbers.