Related papers: On the distance between perfect numbers
In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.
In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or…
We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these…
In this work, we define a triangle area number to be the area number of a triangle whose sides have integer lengths, and whose area is a rational number. In Result 3, on page 17, we prove that every triangle area number is in fact an…
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not…
The Erd\H{o}s-Mollin-Walsh conjecture, asserting the nonexistence of three consecutive powerful integers, remains a celebrated open problem in number theory. A natural line of inquiry, following recent work by Chan (2025), is to investigate…
In this paper, we study universal sums of triangular numbers and squares. Specifically, we prove that a sum of triangular numbers and squares is universal if and only if it represents…
We discuss the existence of the angle between two curves in Teichm\"uller spaces and show that, in any infinite dimensional Teichm\"uller space, there exist infinitely many geodesic triangles each of which has the same three vertices and…
This note concerns the non-existence of three consecutive powerful numbers. We use Pell equations, elliptic curves, and second-order recurrences to show that there are no such triplets with the middle term a perfect cube and each of the…
We proved that there are infinitely many pairs of twin prime.
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and…
For any particularly interesting theorem one proof is never enough. Instead, the first proof sets the challenge to find a more elegant method that illuminates subtle features of the math, is simpler to understand, or even avoids using…
We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many…
For a given integer $m$ and any residue $a \pmod{m}$ that can be written as a sum of 3 squares modulo $m$, we show the existence of infinitely many integers $n \equiv a \pmod{m}$ such that the number of representations of $n$ as a sum of…
This small note proves that the set of triangular numbers is a finitely stable additive basis. This, together with a previous result by the author, shows that triangular numbers and squares are, among all polygonal numbers, the only ones…
In this paper, we show that, for any integers $n\geq 2$ and $g\geq 2$, there exist genus-$g$ Heegaard splittings of compact 3-manifolds with distance exactly $n$.
We call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $\sigma(k)(m+1)=2n$, where $\sigma$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect…
A complete list of irreducible triangulations is identified on the M\"obius band.
This paper provides a link between integer exponentiation and integer tetration since it is devoted to introducing some peculiar sets of perfect powers characterized by any given value of their constant congruence speed, revealing a…
A general construction yielding infinitely many families of $D(m^2)$-triples of triangular numbers is presented. Moreover, each triple obtained from this construction contains the same triangular number $T_n$.