Prime Number and Division

Number Theory Level 2

Positive integers $a,b$ are such that $p=a^2+b^2$ is a prime and $p-5$ divides 8.

Assume there are integers $x,y$ such that $ax^2-by^2$ divides $p$ .

Is the statement below true or false?

$x$ and $y$ divide $p$ , respectively.

True False

1 solution

A Former Brilliant Member
Jul 27, 2020

Since $p$ is prime, $ax^2-by^2$ can be $1,p$ only. Also $p-5$ can be $1,2,4,8\implies p$ can be $7,13$ only. Of these, $7$ can't be expressed as the sum of two squares. So, the only possible value of $p$ is $13=3^2+2^2$ . So $a=3,b=2$ . Of all the combinations of $x$ and $y$ , only $x=1,y=1$ yields $3x^2-2y^2=1$ . Therefore the statement is True .

Prime Number and Division

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