Related papers: Je\'{s}manowicz' conjecture and Fermat numbers

Paul Erdos conjectured that for every n in N, n>1, there exist a, b, c natural numbers, not necessarily distinct, so that 4/n=1/a+1/b+1/c (see \cite{rg}). In this paper we prove an extension of Mordell's theorem and formulate a conjecture…

In 2013, Farhi conjectured that for each $m\geq 3$, every natural number $n$ can be represented as $\lfloor x^2/m\rfloor+\lfloor y^2/m\rfloor+\lfloor z^2/m\rfloor$ with $x,y,z\in\Z$, where $\lfloor\cdot\rfloor$ denotes the floor function.…

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c <…

The generalized Erd\H{o}s-Straus conjecture, proposed by Wac\l{}aw Sierpi\'{n}ski in 1956, asks whether the Diophantine equation \[ \frac{5}{a} = \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \] admits positive integer solutions $b,c,d \in…

Let $F_n$ denote the $n^{th}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_1^p+2F_2^p+\cdots+kF_{k}^p=F_{n}^q$ in the positive integers $k$ and $n$, where $p$ and $q$ are given positive integers.…

We determine all triples $(a,b,n)$ of positive integers such that $a$ and $b$ are relatively prime and $n^k$ divides $a^n + b^n$ (respectively, $a^n - b^n$), when $k$ is the maximum of $a$ and $b$ (in fact, we answer a slightly more general…

[This is an older version of the paper, which will be updated soon.] In the present paper, we continue our research on the generalized Fermat equation $x^r + y^s = z^t$ with signature $(r, s, t)$, where $r, s, t \ge 2$ are positive integers…

A long-standing conjecture asserts that the polynomial \[p(t) = \text{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n \times n$ positive semidefinite Hermitian matrices. The…

While solving a special case of a question of Erd\H{o}s and Graham Steinerberger asks for all integers $n$ with $\phi(n)=\frac{2}{3} \cdot (n+1)$. He discovered the solutions $n\in\{5, 5 \cdot 7, 5\cdot 7\cdot 37, 5\cdot 7\cdot 37\cdot…

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(\lambda)+1/y(\lambda)+1/z(\lambda)$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The…

Every odd prime number p can be written in exactly (p + 1)/2 ways as a sum ab+cd of two ordered products ab and cd such that min(a, b) > max(c, d). An easy corollary is a proof of Fermat's Theorem expressing primes in 1 + 4N as sums of two…

We prove the Ribenboim hypothesis, which states that if, starting from some integer $N$, consecutive prime numbers $p_ {n}$, $p_{n+1}$ satisfy the inequality $\sqrt {p_ {n+1}}-\sqrt{p_{n}} <1$, then the Landau problem # 4 (1912) has a…

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture…

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$…

Let $a,b,c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this paper, by analyzing the gap rule for solutions of the ternary purely exponential diophantine equation $a^x+b^y=c^z$, we prove that if $\max\{a,b,c\}\geq…

A conjecture of Cai-Zhang-Shen for figurate primes says that every integer $k>1$ is the sum of two figurate primes. In this paper we give an equivalent proposition to the conjecture. By considering extreme value problems with constraints…

The non-zero integer solution set is derived for C^n = A^n + B^n. The non-zero integer solution set for n = 2 is [C - (a + b)]^2 = 2ab. The variables a and b equal (C - A) and (C - B) respectively and are nonzero integer factors of 2M^2…

Let $g$ be a map from the set of positive integers into itself defined as follows: Let $x$ be a positive integer. If $x$ is odd, then $g(x)=3x+1$, and if $x$ is even, then $g(x)=x/2$. The $3x+1$ conjecture, also called the Collatz…

Let $S_k(m):=\sum_{j=1}^{m-1}j^k$ denote a power sum. In 2011, Kellner proposed the conjecture that for $m>3$ the ratio $S_k(m+1)/S_k(m)$ is never an integer, or, equivalently, that for any positive integer $a$, the equation $aS_k(m)=m^k$…

Let $m$ be any positive integer and let $\delta_1,\delta_2\in\{1,-1\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\le C_m$ such that $$\left(\frac{p_{n+i}}{p_{n+j}}\right)=\delta_1\…