Let a, b, c be relatively prime positive integers such that $a^2+b^2=c^2$. Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${\left(an\right)}^x+{\left(bn\right)}^y={\left(cn\right)}^z$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${(u^2-v^2)}^x+{\left(2uv\right)}^y={(u^2+v^2)}^z$, for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

Let a,b,c be fixed coprime positive integers with mina,b,c > 1, and let m = maxa,b,c. Using the Gel’fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation $a^x+b^y=c^z$ satisfy maxx,y,z < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with minx,y,z > 1.

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 $n2·{10}^{17}$.

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 $D$ 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...