Question

Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?

score 33.4k · Answer 1 · 2026-04-22 23:20:26

33.4k

Views

Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?

Published on 22 Apr 2026 - 23:20

score 233 · Answer 2 · 2026-05-03 07:14:43

233

Views

Find $n \in N$, for which $a+b+c+d = n \sqrt{(abcd)}$ has a solution $(a, b, c, d)$ in natural numbers.

Published on 03 May 2026 - 7:14

score 487 · Answer 3 · 2026-04-23 22:23:11

487

Views

Find all natural solutions $(a, b)$ such that $(ab - 1) \mid (a^2 + a - 1)^2$.

Published on 23 Apr 2026 - 22:23

score 454 · Answer 4 · 2026-04-25 04:19:46

454

Views

A generalization of the (in)famous IMO 1988 problem 6: If $\frac{a^2 + b^2 - abc}{ab + 1}$ is a positive integer then it is a square.

Published on 25 Apr 2026 - 4:19

score 236 · Answer 5 · 2026-04-05 19:53:13

236

Views

Number Theory Problem Germany 2003

Published on 05 Apr 2026 - 19:53

score 227 · Answer 6 · 2020-03-12 14:43:12

227

Views

Find all positive integer pairs $(a, b)$ such that $(ab + a + b) \mid (a^2 + b^2 + 1)$.

Published on 12 Mar 2020 - 14:43

score 328 · Answer 7 · 2020-03-28 19:06:35

328

Views

Find all positive integers $n$ for which the equation $x + y + u + v = n \sqrt{ xyuv }$ has a solution in positive integers.

Published on 28 Mar 2020 - 19:06

score 209 · Answer 8 · 2026-04-26 22:38:15

209

Views

All integer values of $\frac{a^2+b^2+1}{ab-1}$

Published on 26 Apr 2026 - 22:38

score 204 · Answer 9 · 2026-04-05 14:27:21

204

Views

Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.

Published on 05 Apr 2026 - 14:27

score 138 · Answer 10 · 2026-04-02 10:15:15

138

Views

polynomial equation $ A(x+y_1)(x+y_2)...(x+y_n) + B(x+z_1)(x+z_2)...(x+z_k) = f(x) $ ??

Published on 02 Apr 2026 - 10:15

score 253 · Answer 11 · 2026-04-22 23:22:07

253

Views

Number Theory And Vieta Jumping

Published on 22 Apr 2026 - 23:22

score 481 · Answer 12 · 2026-04-22 23:22:07

481

Views

What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$?

Published on 22 Apr 2026 - 23:22

score 523 · Answer 13 · 2026-04-22 23:20:25

523

Views

Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Published on 22 Apr 2026 - 23:20

score 145 · Answer 14 · 2026-04-22 23:20:40

145

Views

Proving $ \left\{\frac{y^2+z^2+2}{yz}\ \mid\ y,z\in\mathbb{N}\right\}\cap\mathbb{N}=\{4\} $

Published on 22 Apr 2026 - 23:20

score 85 · Answer 15 · 2026-04-22 23:22:12

85

Views

Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?

Published on 22 Apr 2026 - 23:22

List Question

Is it true that $f(x,y)=\frac{x^2+y^2}{xy-t}$ has only finitely many distinct positive integer values with $x$, $y$ positive integers?

Find $n \in N$, for which $a+b+c+d = n \sqrt{(abcd)}$ has a solution $(a, b, c, d)$ in natural numbers.

Find all natural solutions $(a, b)$ such that $(ab - 1) \mid (a^2 + a - 1)^2$.

A generalization of the (in)famous IMO 1988 problem 6: If $\frac{a^2 + b^2 - abc}{ab + 1}$ is a positive integer then it is a square.

Number Theory Problem Germany 2003

Find all positive integer pairs $(a, b)$ such that $(ab + a + b) \mid (a^2 + b^2 + 1)$.

Find all positive integers $n$ for which the equation $x + y + u + v = n \sqrt{ xyuv }$ has a solution in positive integers.

All integer values of $\frac{a^2+b^2+1}{ab-1}$

Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.

polynomial equation $ A(x+y_1)(x+y_2)...(x+y_n) + B(x+z_1)(x+z_2)...(x+z_k) = f(x) $ ??

Number Theory And Vieta Jumping

What are all possible positive integers $k$ such that $k=\frac{a^2+b^2+c^2}{bc+ca+ab}$ for some positive integers $a$, $b$, and $c$?

Find a positive integer solution to $xyzw=504(x^2+y^2+z^2+w^2)$

Proving $ \left\{\frac{y^2+z^2+2}{yz}\ \mid\ y,z\in\mathbb{N}\right\}\cap\mathbb{N}=\{4\} $

Why do we need the minimum solution of (A, B) with respect to some funtion in Vieta jumping?

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