Number of complex solutions

Question

Given the following equation:

$$ x^{259}=1 $$

$$ x^{413}=1 $$

How many complex solutions for x have?

Thanks

I count two equations... Is that a system or two independent equations? — ajotatxe, Dec 09 '14 at 22:36

score 2 · Accepted Answer · answered Dec 09 '14 at 22:39

2

There are seven solutions of the form $e^{i2k\pi/7}$ for $k=0,...,6$, because the common factor of 259 and 413 is 7. Six of them are "complex" and the other one ($k=0$) is just $1$.

answered Dec 09 '14 at 22:39

Suzu Hirose

11,660

"Common factor" being gcd(413,259) – servabat Dec 09 '14 at 22:42
1

$1$ is a perfectly good complex number too – Andrea Mori Dec 09 '14 at 22:45
Yeah, if real numbers were not complex, the complex plane wouldn't be connected. – ajotatxe Dec 09 '14 at 22:47
1

That's why I put "complex" in quotes. – Suzu Hirose Dec 09 '14 at 22:48

score 2 · Answer 2 · answered Dec 09 '14 at 22:42

2

If you mean that $x$ must satisfy both equations, note that $\gcd(259,413)=7$.

Prove now:

Every $7$th root of $1$ satisfies both equations.
Every solution $x$ of both equations must be a $7$th root of $1$. Hint: use Bezout's identity.

answered Dec 09 '14 at 22:42

ajotatxe

65,084

Number of complex solutions

2 Answers2