Related papers: The Erd\"os-Straus Conjecture and Pythagorean Prim…

A strong version of Andrica's conjecture can be formulated as follows: Except for $p_n\in\{3,7,13,23,31,113\}$, that is $n\in\{2,4,6,9,11,30\}$, one has$\sqrt{p_{n+1}}-\sqrt{p_n} < \frac{1}{2}.$ While a proof is far out of reach I shall…

The Markov numbers are the positive integer solutions of the Diophantine equation $x^2 + y^2 + z^2 = 3xyz$. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and…

Let $u_n$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_1,p_2,\dots,p_s$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $u_n+u_m=w…

Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm…

We study solvability of the Diophantine equation \begin{equation*} \frac{n}{2^{n}}=\sum_{i=1}^{k}\frac{a_{i}}{2^{a_{i}}}, \end{equation*} in integers $n, k, a_{1},\ldots, a_{k}$ satisfying the conditions $k\geq 2$ and $a_{i}<a_{i+1}$ for…

A brief history and two formulations of the Diophantine problem's requirements are presented. One tier consisting of three two-parameter solutions is studied for its ability to provide examples for the small natural numbers considered.…

Let (a,b,c)be a Pythagorean triple with c being the hypotenuse length, and h being the altitude to the hypotenuse. Also, let v,k,l be positive integers with k and l being relatively prime.We say(Definition1 in this work)that the Pythagorean…

Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants $\lambda_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\,…

In this note we investigate the problem of finding pairs of Pythagorean triangles $(a, b, c), (A, B, C)$, with given catheti ratios $A/a, B/b$. In particular, we prove that there are infinitely many essentially different ("non-similar")…

We study the Constraint Satisfaction Problem CSP(A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

In the fall 2011 issue of the Journal'Mathematics and Computer Education', author Unal Hasan, in the one page article "Proof without Words", gives a purely geometric proof of the equality, arctan(1/3)+ arctan(1/7) = arctan(1/2) (1) (See…

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p…

Let $1<k<14/5$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

In this paper, we consider the exponential Diophantine equation \( (2^k-1)(b^k-1)=y^q \) with $k\ge 2$, odd integer $b$ and an odd prime exponent $q$ and obtain effective upper bounds for $q$ in terms of $b$. In particular, we show that…

If $p_k$ is the k-th prime, the Firoozbakht conjecture states that the sequence $(p_k)^{1/k}$ is strictly decreasing. We use the table of first-occurrence prime gaps in combination with known bounds for the prime-counting function to verify…

Let $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

We prove that for given integers b and c, the diophantine equation x^2+bx+c=y^2, has finitely many integer solutions(i.e. pairs in ZxZ),in fact an even number of such solutions(including the zero or no solutions case).We also offer an…

The goal of the present paper is to present a method of proving of Diophantine inequalities with primes through the use of auxiliary inequalities and available evaluations of the difference between consecutive primes. We study the Legendre…

We present some new ideas on important problems related to primes. The topics of our discussion are: simple formulae for primes, twin primes, Sophie Germain primes, prime tuples less than or equal to a predefined number, and their…

Let $k$ be a positive integer, and let $a,b$ be coprime positive integers with $\min\{a,b\}>1$. In this paper, using a combination of some elementary number theory techniques with classical results on the Nagell-Ljunggren equation, the…