summary:The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \Bbb N$, satisfy the equation $x^y+y^x=z^z$. In this paper it is shown that the same equation has no integer solution with $\min \{x,y,z\} > 1$, thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed
International audienceThis paper gives a complete four-parameter solution of the simultaneous diopha...
summary:Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1...
summary:Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$,...
summary:The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \Bbb N$, satisfy the e...
summary:In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New ...
[[abstract]]In this paper, we discuss the positive integers solutions of the Diophantine equations x...
If a, b and n are positive integers with b = a and n = 3, then the equation of the title possesses a...
In this paper, we show that (3, 0, 3) is a unique non-negative integer solution for the Diophantine ...
summary:Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $...
summary:For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive in...
The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z $ wi...
{ Beal's Conjecture :} The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive...
Given two positive integers $a,b$, we discuss the diophantine equation $f(a^m,y)=b^n$, to be solved ...
summary:Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respect...
For all integers a, b, c, assuming the true abc n−conjecture in the case n = 5, there are finitely m...
International audienceThis paper gives a complete four-parameter solution of the simultaneous diopha...
summary:Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1...
summary:Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$,...
summary:The triples $(x,y,z)=(1,z^z-1,z)$, $(x,y,z)=(z^z-1,1,z)$, where $z\in \Bbb N$, satisfy the e...
summary:In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New ...
[[abstract]]In this paper, we discuss the positive integers solutions of the Diophantine equations x...
If a, b and n are positive integers with b = a and n = 3, then the equation of the title possesses a...
In this paper, we show that (3, 0, 3) is a unique non-negative integer solution for the Diophantine ...
summary:Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2\nmid x$, $...
summary:For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive in...
The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive integers $x, y, z $ wi...
{ Beal's Conjecture :} The equation $z^\xi=x^\mu+y^\nu$ has no solution in relatively prime positive...
Given two positive integers $a,b$, we discuss the diophantine equation $f(a^m,y)=b^n$, to be solved ...
summary:Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respect...
For all integers a, b, c, assuming the true abc n−conjecture in the case n = 5, there are finitely m...
International audienceThis paper gives a complete four-parameter solution of the simultaneous diopha...
summary:Let $a$, $b$, $c$, $r$ be positive integers such that $a^{2}+b^{2}=c^{r}$, $\min (a,b,c,r)>1...
summary:Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$,...
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