April 6, 2012
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September 29, 2023
Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square times a single prime unless $\alpha = 1$. It is a direct consequence of a certain property of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where $l$ denotes on...
December 3, 2015
Euler showed that if an odd perfect number exists, it must be of the form $N = p^\alpha q_{1}^{2\beta_{1}}$ $\ldots$ $q_{k}^{2\beta_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $\alpha$, $\beta_{i} \geq 1$, for $1 \leq i \leq k$, with $p \equiv \alpha \equiv 1 \pmod{4}$. In 2005, Evans and Pearlman showed that $N$ is not perfect, if $3|N$ or $7|N$ and each $\beta_{i} \equiv 2 \pmod{5}$. We improve on this result by removing the hypothesis that $3|N$ or $7|N$ ...
January 20, 2020
Let $n$ and $k$ be positive integers and $\sigma(n)$ 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 $\sigma (n)=2n-(d_1+d_2+\ldots + d_k)$. In this article, we show that the only odd exactly $3$-deficient-perfect number with at most two distinct prime factors is $1521=3^2 \cdot 13^2$.
January 18, 2018
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 ...
March 4, 2008
Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.
February 8, 2020
In this paper, we prove the conjecture that if there is an odd perfect number, then there are infinitely many of them.
February 26, 2020
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 $\sigma(m^2) \equiv 1 \pmod 4$ holds if and only if $p \equiv k \pmod 8$. We then give an application of this result to the case when $\sigma(m^2)/p^k$ is a square.
May 25, 2020
Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $\sigma(n!+1)$, $\sigma(2^n+1)$, and related sequences.
February 10, 2022
Let $q^k n^2$ be an odd perfect number with special prime $q$. Define the GCDs $$G = \gcd\bigg(\sigma(q^k),\sigma(n^2)\bigg)$$ $$H = \gcd\bigg(n^2,\sigma(n^2)\bigg)$$ and $$I = \gcd\bigg(n,\sigma(n^2)\bigg).$$ 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 $\sigma(q^k)/2, n,$ and $\gcd\bigg(\sigma(q^k)/2,n\bigg)$. Afterwards, we prove that if $G = H = I$,...
October 31, 2018
Let $N$ be an odd perfect number. Let $\omega(N)$ be the number of distinct prime factors of $N$ and let $\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \frac{302}{113}\omega - \frac{286}{113} \leq \Omega. $ If $3|N$, then $\frac{66}{25}\omega-5\leq\Omega.$ 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 $\omega(N)$ in terms ...