Related papers: Note on the Theory of Perfect Numbers

It is conjectured that for a perfect number $m,$ $\rm{rad}(m)\ll m^{\frac{1}{2}}.$ We prove bounds on the radical of multiperfect number $m$ depending on its abundancy index. Assuming the ABC conjecture, we apply this result to study gaps…

There has been always an ambiguity in division when zero is present in the denominator. So far this ambiguity has been neglected by assuming that division by zero as a non-allowed operation. In this paper, I have derived the new set of…

We extend our previous work on odd spoof multiperfect numbers to the case where spoof factor multiplicities exceed $2$. This leads to the identification of $11$ new integers that would be odd multiperfect numbers if one of their prime…

Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related…

In this note, we fix a gap in a proof of the first author that 28 is the only even perfect number which is the sum of two perfect cubes. We also discuss the situation for higher powers.

A new problem is studied, the concept of exactness of a second order nonlinear ordinary differential equations is established. A method is constructed to reduce this class into a first order equations. If the second order equation is not…

The problem as to when two noncommuting observables are considered to have the same value arises commonly, but shows a nontrivial difficulty. Here, an answer is given by establishing the notion of perfect correlations between noncommuting…

A classical theorem of Kempner states that the sum of the reciprocals of positive integers with missing decimal digits converges. This result is extended to much larger families of "missing digits" sets of positive integers with convergent…

A permutiple is the product of a digit preserving multiplication, that is, a number which is an integer multiple of some permutation of its digits. Certain permutiple problems, particularly transposable, cyclic, and, more recently,…

In this note, we present some new results on even almost perfect numbers which are not powers of two. In particular, we show that $2^{r+1} < b$, if ${2^r}{b^2}$ is an even almost perfect number.

In this article, based on ideas and results by J. S\'andor (2001, 2004), we define $k$-multiplicatively $e$-perfect numbers and $k$-multiplicatively $e$-superperfect numbers and prove some results on them. We also characterize the…

In the paper we can prove that every integer can be written as the sum of two integers, one perfect square and one squarefree. We also establish the asympotic formula for the number of representations of an integer in this form. The result…

We say a natural number~$n$ is abundant if $\sigma(n)>2n$, where $\sigma(n)$ denotes the sum of the divisors of~$n$. The aliquot parts of~$n$ are those divisors less than~$n$, and we say that an abundant number~$n$ is pseudoperfect if there…

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…

Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, 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…

The equality between the number of odd spin structures on a Riemann surface of genus g, with $2^g - 1$ being a Mersenne prime, and the even perfect numbers, is an indication that the action of the modular group on the set of spin structures…

The fibbinary numbers are positive integers whose binary representation contains no consecutive ones. We prove the following result: If the $j$th odd fibbinary is the $n$th \emph{odd} fibbinary number, then $j = \lfloor n\phi^2 \rfloor -…

We introduce a new -as far as we know- problem, according to which we are asked to match sequences of two digits in matrices having entries among those two digits (but others too) and prove that this problem is NP-complete

Let $b$ be a numeration base. A $b$-Niven number is one that is divisible by the sum of its base $b$ digits. We introduce high degree $b$-Niven numbers. These are $b$-Niven numbers that have a power greater than $1$ that is $b$-Niven…

We give a possible explanation for the mystery of a missing number in the statement of a problem that asks for the non-negative integers to be partitioned into three subsets. We interpret the missing number as one of the clues that can lead…