Related papers: Solving the Odd Perfect Number Problem: Some Old a…

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 =…

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…

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…

A unitary perfect number is a positive integer n satisfying \sigma^*(n)=2n, where \sigma^* sums unitary divisors. Only five examples are known, and no sixth has been found. We revisit the Subbarao-Warren problem by keeping the seed factor…

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same…

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…

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…

For $n\leq 1.5 \cdot 10^{10}$, we have found a total number of 1268 solutions to the Erd\"os-Sierpi\'nski problem finding positive integer solutions of $\sigma(n)=\sigma(n+1)$, where $\sigma(n)$ is the sum of the positive divisors of n. On…

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}…

In his classic text, \emph{Combinatory Analysis}, MacMahon defined a perfect partition of a positive integer $n$ as a partition whose parts contain exactly one partition of every positive integer not exceeding $n$. In this paper we apply…

In this paper, I proved that $$N=p_1+p_2+2p_3, p_1\sim N/2, p_2\sim N/2, p_3=o(N),$$ where $N$ is a large even number, and $p_i\ (i=1,2,3)$ are odd primes.

We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain…

Let $\sigma(n)$ be the sum of the divisors of $n$. Kalita and Saikia defined a number $n$ to be near superperfect if $2n+d=\sigma(\sigma(n))$ for some positive divisor $d$ of $n$. We extend some of their results about near superperfect…

We shall prove that if $N=p^\alpha q_1^{2\beta_1} q_2^{2\beta_2} \cdots q_{r-1}^{2\beta_{r-1}}$ is an odd perfect number such that $p, q_1, \ldots, q_{r-1}$ are distinct primes, $p\equiv\alpha\equiv 1\mod{4}$ and $t$ divides $2\beta_i+1$…

The divisor function $\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\sigma(n) - n > n$ and perfect when $\sigma(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor…

We study the equation $F_n + F_m = y^p$, where $F_n$ and $F_m$ are respectively the $n$-th and $m$-th Fibonacci numbers and $p \ge 2$. We find all solutions under the assumption $n \equiv m \pmod{2}$.

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…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or…

In 1960, W. Sierpinski proved that there are infinitely many positive odd numbers $k$, such that for any positive integer $n$, $k\times2^n+1$ is a composite number. Such numbers are called "Sierpinski numbers". In this study, by using…