summary:In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = \lowercase {n!} z^{n}$ has no integer solutions with $n\in \mathbb {N_{+}}$ and $n>2$. But, contrary to this expectation, we show that for $n = 3$, this equation has infinitely many primitive integer solutions, i.e.~the solutions satisfying the condition $\gcd (x, y, z)=1$
In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the eq...
Using only elementary arguments, Cassels solved the Diophantine equation (x−1)3+x3+(x+1)3=z2 in inte...
summary:Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respect...
summary:In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New ...
In p. 219 of R.K. Guy's Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we a...
We consider the Brocard-Ramanujan type diophantine equation $y^2=x!+A$ and ask about values of $A\in...
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:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solution...
In this note, we show that the ABC-conjecture implies that a diophantine equation of the form P(x) =...
summary:For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive in...
International audienceA nontrivial solution of the equation A!B! = C! is a triple of positive intege...
We prove that the equation (-l)(n-1)/2 (((n-1)/2)!)2+an-1=nk in positive integers n, a, k with n > 2...
If a, b and n are positive integers with b = a and n = 3, then the equation of the title possesses a...
summary:Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$,...
In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the eq...
Using only elementary arguments, Cassels solved the Diophantine equation (x−1)3+x3+(x+1)3=z2 in inte...
summary:Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respect...
summary:In p.~219 of R.K. Guy's \emph {Unsolved Problems in Number Theory}, 3rd edn., Springer, New ...
In p. 219 of R.K. Guy's Unsolved Problems in Number Theory, 3rd edn., Springer, New York, 2004, we a...
We consider the Brocard-Ramanujan type diophantine equation $y^2=x!+A$ and ask about values of $A\in...
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:We study the Diophantine equations $(k!)^n -k^n = (n!)^k-n^k$ and $(k!)^n +k^n = (n!)^k +n^k...
Erd\"os and Obl\'ath proved that the equation $n!\pm m!=x^p$ has only finitely many integer solution...
In this note, we show that the ABC-conjecture implies that a diophantine equation of the form P(x) =...
summary:For any positive integer $D$ which is not a square, let $(u_1,v_1)$ be the least positive in...
International audienceA nontrivial solution of the equation A!B! = C! is a triple of positive intege...
We prove that the equation (-l)(n-1)/2 (((n-1)/2)!)2+an-1=nk in positive integers n, a, k with n > 2...
If a, b and n are positive integers with b = a and n = 3, then the equation of the title possesses a...
summary:Consider the equation in the title in unknown integers $(x,y,k,l,n)$ with $x \ge 1$, $y >1$,...
In 1876 Brocard, and independently in 1913 Ramanujan, asked to find all integer solutions for the eq...
Using only elementary arguments, Cassels solved the Diophantine equation (x−1)3+x3+(x+1)3=z2 in inte...
summary:Let $\mathbb {Z}$, $ \mathbb {N}$ be the sets of all integers and positive integers, respect...
DOI
Add to ORKG