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Acta Arithmetica

### A note on a conjecture of Jeśmanowicz

Colloquium Mathematicae

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 ${\left({u}^{2}-{v}^{2}\right)}^{x}+{\left(2uv\right)}^{y}={\left({u}^{2}+{v}^{2}\right)}^{z}$, 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.

Acta Arithmetica

### A note on Jeśmanowicz' conjecture

Colloquium Mathematicae

Acta Arithmetica

Acta Arithmetica

### A note on ternary purely exponential diophantine equations

Acta Arithmetica

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.

Acta Arithmetica

Acta Arithmetica

### A note on the diophantine equation $\left(\genfrac{}{}{0pt}{}{k}{2}\right)-1={q}^{n}+1$

Colloquium Mathematicae

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 $\left(k,q,n\right)$. Moreover, all solutions $\left(k,q,n\right)$ satisfy $k{10}^{{10}^{182}}$, $q{10}^{{10}^{165}}$ and $n2·{10}^{17}$.

Acta Arithmetica

### A note on the diophantine equation ${x}^{2}+{b}^{Y}={c}^{z}$

Czechoslovak Mathematical Journal

Let $a$, $b$, $c$, $r$ be positive integers such that ${a}^{2}+{b}^{2}={c}^{r}$, $min\left(a,b,c,r\right)>1$, $gcd\left(a,b\right)=1,a$ 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 $\left(x,y,z\right)=\left(a,2,r\right)$ with $min\left(y,z\right)>1$.

Acta Arithmetica

Acta Arithmetica

### A Note on the Diophantine Equation x2 + 4D = yp.

Monatshefte für Mathematik

Acta Arithmetica

### A note on the exponential Diophantine equation ${\left(4m²+1\right)}^{x}+{\left(5m²-1\right)}^{y}={\left(3m\right)}^{z}$

Colloquium Mathematicae

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 ${\left(4m²+1\right)}^{x}+{\left(5m²-1\right)}^{y}={\left(3m\right)}^{z}$ has only the positive integer solution (x,y,z) = (1,1,2).

Acta Arithmetica

### A note on the number of solutions of the generalized Ramanujan-Nagell equation ${x}^{2}-D={p}^{n}$

Czechoslovak Mathematical Journal

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 $N\left(D,p\right)$, and also prove that if the equation ${U}^{2}-D{V}^{2}=-1$ has integer solutions $\left(U,V\right)$, the least solution $\left({u}_{1},{v}_{1}\right)$ 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...

### A rationality condition for the existence of odd perfect numbers.

International Journal of Mathematics and Mathematical Sciences

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