### A conjecture concerning the exponential diophantine equation ${a}^{x}+{b}^{y}={c}^{z}$

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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 $n2\xb7{10}^{17}$.

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 $b\equiv 3\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}4\right)$ 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 ${(4m\xb2+1)}^{x}+{(5m\xb2-1)}^{y}={\left(3m\right)}^{z}$ 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...