Related papers: On practical numbers of some special forms
A positive integer $n$ is said to be a practical number if every integer in $[1,n]$ can be represented as the sum of distinct divisors of $n$. In this article, we consider practical numbers of a given polynomial form. We give a necessary…
Practical numbers are positive integers $n$ such that every positive integer less than or equal to $n$ can be written as a sum of distinct positive divisors of $n$. In this paper, we show that all positive integers can be written as a sum…
A "practical number" is a positive integer $n$ such that every positive integer less than $n$ can be written as a sum of distinct divisors of $n$. We prove that most of the binomial coefficients are practical numbers. Precisely, letting…
A positive integer $n$ is called practical if all integers between $1$ and $n$ can be written as a sum of distinct divisors of $n$. We give an asymptotic estimate for the number of integers $\le x$ which have a practical divisor $\ge y$.
We study triples {a,b,c} of distinct nonzero rational numbers such that a+1,b+1,c+1,ab+1,ac+1,bc+1 and abc+1 are all perfect squares. We prove that there exist infinitely many such triples. In contrast, we show that no triple of positive…
We show that for infinitely many square-free integers q there exist infinitely many triples of rational numbers {a, b, c} such that a^2 + q, b^2 + q, c^2 + q, ab + q, ac + q and bc + q are squares of rational numbers.
A number $n$ is practical if every integer in $[1,n]$ can be expressed as a subset sum of the positive divisors of $n$. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and…
An integer $n$ is called practical if every $m\le n$ can be written as a sum of distinct divisors of $n$. We show that the number of practical numbers below $x$ is asymptotic to $c x/\log x$, as conjectured by Margenstern. We also give an…
A practical number is a positive integer $n$ such that all positive integers less than $n$ can be written as a sum of distinct divisors of $n$. Leonetti and Sanna proved that, as $x \to +\infty$, the central binomial coefficient…
We show that there are infinitely many triples of positive integers a, b, c (greater than 1) such that ab + 1, ac + 1, bc + 1 and abc + 1 are all perfect squares.
Let $A$ be a set of positive integers. We define a positive integer $n$ as an $A$-practical number if every positive integer from the set $\left\{1,\ldots ,\sum_{d\in A, d\mid n}d\right\}$ can be written as a sum of distinct divisors of $n$…
The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $\epsilon > 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c…
This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be…
An integer $n\ge 1$ is said to be practical if every natural number $ m \le n$ can be expressed as a sum of distinct positive divisors of $n$. The number of practical numbers up to $x$ is asymptotic to $c x/\log x$, where $c$ is a constant.…
We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
By an $abc$ triple, we mean a triple $(a,b,c)$ of relatively prime positive integers $a,b,$ and $c$ such that $a+b=c$ and $\operatorname{rad}(abc)<c$, where $\operatorname{rad}(n)$ denotes the product of the distinct prime factors of $n$.…
A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and…
We prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally…
This paper extends the Pythagorean Theorem to positive and negative real exponents to take the form a^n + b^n = c^n and makes use of the definition gamma = b/a >= 1. For the case of n in the set of positive real numbers, n greater than or…