A number theory problem by ritwik jain

Number Theory Level 3

True or False?

If $a$ and $b$ are positive integers such that $ab + 1$ divides $a^2 + b^2$ then $\dfrac {a^2+b^2}{ab+1}$ is a perfect square.

False True

1 solution

Tapas Mazumdar
May 17, 2017

Vieta root jumping is the best way to solve this problem. The solution has already been discussed here .

Even putting $a=b=1$ will prove

Md Zuhair - 4 years ago

Finding a specific case doesn't mean that the statement is always true. E.g. "If $a$ and $b$ are positive integers such that $a + b \mid a^2 - b^2$ , then $a-b$ is a perfect square. The statement is true for $a = b = 1$ , but doesn't mean it is always true.

Calvin Lin Staff - 4 years ago

Ok. Thank you sir

Md Zuhair - 4 years ago

A number theory problem by ritwik jain

1 solution

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