Related papers: On certain ratios regarding integer numbers which …

Some finite series of harmonic numbers involving certain reciprocals are evaluated. Products of such reciprocals are expanded in a sum of the individual reciprocals, leading to a computer program. A list of examples is provided.

This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…

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 survey paper, I first review the history of Bernoulli numbers, then examine the modern definition of Bernoulli numbers and the appearance of Bernoulli numbers in expansion of functions. I revisit some properties of Bernoulli numbers…

In this article, we discuss whether a single congruent number $t$ can have two (or more) distinct triangles with the same hypotenuse. We also describe and carry out computational experimentation providing evidence that this does not occur.

The nearest integer continued fraction of a real number $x$ from $[-1/2, 1/2)$ is defined. Some metrical properties of these expansions are presented. We define the approximation coefficients and give an important result on them. The main…

Sums of the form $\sum_{N_m=q}^{n}{\cdots \sum_{N_1=q}^{N_2}{a_{(m);N_m}\cdots a_{(1);N_1}}}$ where the $a_{(k);N_k}$'s are same or distinct sequences appear quite often in mathematics. We will refer to them as recurrent sums. In this…

This paper is concerned with the problem of finding two sets of integers, $\{a_1, a_2, \ldots$, $a_m\}$ and $\{b_1, b_2, \ldots, b_n\}$, such that all the $mn$ sums $a_i+b_j, i=1, \ldots, m, j=1, \ldots, n$, are perfect squares. A method is…

Let $S_k(m):=1^k+2^k+\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that…

Let $p$ and $q$ be two distinct fixed prime numbers and $(n_i)_{i\geq 0}$ the sequence of consecutive integers of the form $p^a\cdot q^b$ with $a,b\ge 0$. Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size…

We determine all integers $n$ such that $n^2$ has at most three base-$q$ digits for $q \in \{2, 3, 4, 5, 8, 16 \}$. More generally, we show that all solutions to equations of the shape $$ Y^2 = t^2 + M \cdot q^m + N \cdot q^n, $$ where $q$…

We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.

We study some divisibility properties of quasiperfect numbers. We show that if $N=(p_1 p_2 \cdots p_t)^{2a}=m^2$ is quasiperfect, then $2a+1$ is divisible by $3$ and $N$ has at least one prime factor smaller than $\exp 716.7944$. Moreover,…

We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of…

We study odd numbers through a straightforward indexing. We focus in particular on odd prime and composite numbers and their distribution. With a counting argument, we calculate the limit of two sums and compare their convergence rate.

The set of short intervals between consecutive primes squared has the pleasant---but seemingly unexploited---property that each interval $s_k:=\{p_k^2, \dots,p_{k+1}^2-1\}$ is fully sieved by the $k$ first primes. Here we take advantage of…

Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\lfloor…

We define S(um)anD(ifference) numbers as ordered pairs $(m,\, m+\Delta)$ such that the digital-sum $DS(m(m+\Delta))=\Delta.$ We consider both the decimal and the binary case. If both $m$ and $m+\Delta$ are prime numbers, we refer to SanD…

The theory of Gromov-Hausdorff convergence is applied to sequences of quotient rings of integers. It is shown the existence of limit rings (fields) as the Gromov-Hausdorff limits of sequences of metric quotient rings. The relation of these…

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…