相关论文: On Amicable Numbers With Different Parity

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $\sigma(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems…

We say a natural number $n$ is matchable if there is a bijection from the set of $\tau(n)$ divisors of $n$ to the set $\{1,2,\dots,\tau(n)\}$, where corresponding numbers are relatively prime. We show that the set of matchable numbers has…

A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect…

We give a complete characterization of the parity of $b_8(n)$, the number of $8$-regular partitions of $n$. Namely, we prove that $b_8(n)$ is odd or even depending on whether or not we have the factorisation $24n+7=p^{4a+1}m^2$, for some…

We recall the Alon-Tarsi conjecture on the number of even latin squares. We introduce a map which switches the parity of a latin square under certain requirements. An example is included.

We prove that every odd number $N$ greater than 1 can be expressed as the sum of at most five primes, improving the result of Ramar\'e that every even natural number can be expressed as the sum of at most six primes. We follow the circle…

The aim of this note is to introduce fastest new general methods for the construction of double and single even order magic squares. As in [5], the method for double even order magic squares is fairly straight-forward but some adjustments…

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…

Mersenne numbers and Fermat numbers are two hot and difficult issues in number theory. This paper constructs a special group for every positive odd number other than 1, and discovers an algorithm for determining the multiplicative order of…

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.

We show that there are infinitely many square numbers , which are constrocted by putting two square numbers together , that non of them are divisible by $10$ . We also investigate the interesting properties of some square numbers.

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.

We generalize the definition of spoof perfect numbers to multiperfect numbers and study their characteristics. As a result, we find several new odd spoof multiperfect numbers, akin to Descartes' number. An example is $8999757$, which would…

A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.

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…

We investigate the consecutive primes $p$ and $q$ ($p > q$) for which there exists a pair of natural numbers $(x,y)$ such that $p^x-q^y$ is a perfect square and make some conjectures.

In this paper, we consider the set of partitions $pend(n)$ which enumerates the number of partitions of $n$ wherein the even parts are not allowed to be distinct. Using a result of Newman, we prove a few infinite families of congruences…

An amicable pair for an elliptic curve E/Q is a pair of primes (p,q) of good reduction for E satisfying #E(F_p) = q and #E(F_q) = p. In this paper we study elliptic amicable pairs and analogously defined longer elliptic aliquot cycles. We…

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…

Let p1, p2,..., pn be distinct prime numbers, and let Nn be their product. We prove that, for any positive integer L that is divisible by the least common multiple of p1 minus one, p2 minus one, and so on, and for integers a1, a2,..., an…