We show that the Diophantine equation of the title has, for $n\>1$, no solution in coprime nonzero integers $x,y$ and $z$. Our proof relies upon Frey curves and related results on the modularity of Galois representations.

Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are nonzero real numbers, not all negative, $\delta >0$, $𝒱$ is a well-spaced set, and the ratio ${\lambda }_1/{\lambda }_2$ is algebraic and irrational. Denote by $E(𝒱,N,\delta )$ the number of $v\in 𝒱$ with $v\le N$ such that the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^4+{\lambda }_4{p}_4^5-v|<v^{-\delta }$$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$E(𝒱,N,\delta )\ll N^{1+2\delta -1/72+\epsilon }$$ for any $\epsilon >0$.

Let $n$ be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if $n>1$ and both $6n^2-1$ and $12n^2+1$ are odd primes, then the general elliptic curve $y^2=x^3+(36n^2-9)x-2(36n^2-5)$ has only the integral point $(x,y)=(2,0)$. By this result we can get that the above elliptic curve has only the trivial integral point for $n=3,13,17$ etc. Thus it can be seen that the elliptic curve $y^2=x^3+27x-62$ really is an unusual elliptic curve which has large integral points.

We consider an irreducible curve $𝒞$ in $E^n$, where $E$ is an elliptic curve and $𝒞$ and $E$ are both defined over $\overline{\mathbb{Q}}$. Assuming that $𝒞$ is not contained in any translate of a proper algebraic subgroup of $E^n$, we show that the points of the union $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, where $A$ ranges over all proper algebraic subgroups of $E^n$, form a set of bounded canonical height. Furthermore, if $E$ has Complex Multiplication then the set $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, for $A$ ranging over all algebraic subgroups of $E^n$ of codimension at least $2$, is finite. If $E$ has no Complex Multiplication...

We study the Ljunggren equation Y² + 1 = 2X⁴ using the "multiplication by 2" method of Chabauty.