Related papers: Unitary super 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…

We prove that nine-dimensional exceptional quotient singularities exist.

A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 < 1 +…

Let $\mathrm{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\mathrm{pod}_{-4}(n)$ involving the following infinite family of…

Let $k>2$ be a prime such that $2^k-1$ is a Mersenne prime. Let $n = 2^{\alpha-1}p$, where $\alpha>1$ and $p<3\cdot 2^{\alpha-1}-1$ is an odd prime. Continuing the work of Cai et al. and Jiang, we prove that $n\ |\ \sigma_k(n)$ if and only…

Let U and V be two Bertrand numeration systems, and, a and b the two Parry numbers there are naturally associated with. Suppose they are multiplicatively independent. We prove that, if E is a subset of positive integers which is both U and…

We give all non splitting bi-unitary perfect polynomials over the prime field of two elements, which have only Mersenne polynomials as odd irreducible divisors.

A positive integer $n$ is said to be a Zumkeller number or an integer-perfect number if the set of its positive divisors can be partitioned into two subsets of equal sums. In this paper, we prove several results regarding Zumkeller numbers.…

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper we classify these rings and show that all finite rings of order $p^n$ for $n < 5$ and some of…

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…

We give a short proof of the well-known fact that the unit interval [0,1] is uncountable by means of a simple infinite game. We also show using this game that a (non-empty) perfect subset of [0,1] must be uncountable.

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1…

In this paper we provide a straightforward proof that if a pair of amicable numbers with different parity exists (one number odd and the other one even), then the odd amicable number must be a perfect square, while the even amicable number…

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic…

We prove the existence of a $(d-2)$-dimensional purely unrectifiable set upon which a family of \emph{even} singular integral operators is bounded.

Let $1 < c < 24/19$. We show that the number of integers $n \le N$ that cannot be written as $[p_1^c] + [p_2^c]$ ($p_1$, $p_2$ primes) is $O(N^{1-\sigma+\varepsilon})$. Here $\sigma$ is a positive function of $c$ (given explicitly) and…

In this paper, we define the deformed Euler $(s,t)$-numbers ${\rm e}_{s,t,u}$ Furthermore, we prove that ${\rm e}_{as,a^2t,u^{-1}}$ and ${\rm e}_{as,a^2t,u^{-1}}^{-1}$ are irrational numbers when $a,u\in\mathbb{Q}$ and $\vert au\vert>1$,…

For an odd quadratic space $V$ of Witt index $\geq 3$ over a commutative ring with pseudoinvolution, we classify the subgroups of the odd unitary group $U(V)$ that are normalized by the elementary subgroup $EU_{(e_1,e_{-1})}(V)$ defined by…

We compute all irregular primes less than 163,577,356. For all of these primes we verify that the Kummer-Vandiver conjecture holds and that the lambda-invariant is equal to the index of irregularity.

Let $s(n)$ denote the sum of digits in the binary expansion of the integer $n$. Hare, Laishram and Stoll (2011) studied the number of odd integers such that $s(n)=s(n^2)=k$, for a given integer $k\geq 1$. The remaining cases that could not…