In this note we prove that the equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1=q^n+1$, $q\ge 2,n\ge 3$, has only finitely many positive integer solutions $(k,q,n)$. Moreover, all solutions $(k,q,n)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2·{10}^{17}$.

We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and $P\left(z\right)=a_d{z}^d+a_{d-3}{z}^{d-3}+\cdots +a₁x+a₀$.

Let $a$, $b$, $c$, $r$ be positive integers such that $a^2+b^2=c^r$, $min(a,b,c,r)>1$, $gcd(a,b)=1,a$ is even and $r$ is odd. In this paper we prove that if $b\equiv 3\left(mod4\right)$ and either $b$ or $c$ is an odd prime power, then the equation $x^2+b^y=c^z$ has only the positive integer solution $(x,y,z)=(a,2,r)$ with $min(y,z)>1$.

Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation ${(4m²+1)}^x+{(5m²-1)}^y={\left(3m\right)}^z$ has only the positive integer solution (x,y,z) = (1,1,2).

Let $(a_1,\cdots ,a_m)$ be an $m$-tuple of positive, pairwise distinct integers. If for all $1\le i<j\le m$ the prime divisors of $a_i{a}_j+1$ come from the same fixed set $S$, then we call the $m$-tuple $S$-Diophantine. In this note we estimate the number of $S$-Diophantine quadruples in terms of $\left|S\right|=r$.

Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D,p)$, and also prove that if the equation $U^2-D{V}^2=-1$ has integer solutions $(U,V)$, the least solution $(u_1,v_1)$ of the equation $u^2-p{v}^2=1$ satisfies $p\nmid v_1$, and $D>C\left(p\right)$, where $C\left(p\right)$ is an effectively computable constant...