Find all pairs $(a,b)$ of positive integers for which $a^2-b$ divide $b^2+a$, and $b^2-a$
Divides $a^2 + b$
Search found 47 matches
- Sun May 21, 2017 2:46 am
- Forum: Number Theory
- Topic: Particular divisibility..
- Replies: 1
- Views: 2368
- Sun May 21, 2017 2:44 am
- Forum: Number Theory
- Topic: Determine n
- Replies: 1
- Views: 3405
Determine n
Determine if there are integers $n$ so that $n^2-4$ has exactly $10$ divisors (positive).
- Sun May 21, 2017 2:42 am
- Forum: Algebra
- Topic: Inequality with abc = 1
- Replies: 3
- Views: 7464
Re: Inequality with abc = 1
Good solution Atonu!
- Mon May 15, 2017 2:03 am
- Forum: Algebra
- Topic: Inequality with abc = 1
- Replies: 3
- Views: 7464
Inequality with abc = 1
Let $ x, y, z$ be positive real numbers so that $ xyz = 1$. Prove that
\[ \left( x - 1 + \frac 1y \right) \left( y - 1 + \frac 1z \right) \left( z - 1 + \frac 1x \right) \leq 1.
\]
\[ \left( x - 1 + \frac 1y \right) \left( y - 1 + \frac 1z \right) \left( z - 1 + \frac 1x \right) \leq 1.
\]
- Mon May 15, 2017 1:53 am
- Forum: Number Theory
- Topic: A strange divisibility
- Replies: 1
- Views: 2384
A strange divisibility
Determine all pairs of positive integers $(x,y)$ such that \[ \dfrac{x^2}{2xy^2-y^3+1} \] is a positive integer.
- Sat May 13, 2017 4:24 pm
- Forum: Number Theory
- Topic: Divisibility with a and b
- Replies: 2
- Views: 2852
Re: Divisibility with a and b
Very simple and clear! Thanks
- Sat May 06, 2017 10:12 am
- Forum: Number Theory
- Topic: Divisibility with a and b
- Replies: 2
- Views: 2852
Divisibility with a and b
Determine all ordered pairs $(a,b)$ of positive integers for which $\dfrac{b^3-1}{ab-1}$ is an integer.