Calculate Jacobi symbol

Asked May 15 '20 at 18:18

Active May 15 '20 at 18:18

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Calculate the Jacobi symbol $\left(\frac{n^4+n^2+1}{2n^2+1} \right)$ for every integer $n>0$.

Using the properties of the Jacobi symbol,

$$\left(\frac{n^4+n^2+1}{2n^2+1} \right)=\left(\frac{2n^2+1}{n^2-n+1} \right)\left(\frac{2n^2+1}{n^2+n+1} \right)=\left(\frac{2n-1}{n^2-n+1} \right)\left(\frac{2n-1}{n^2+n+1} \right).$$

However I don't see how this brings me any further. I am clueless, every hint is appreciated

asked May 15 '20 at 18:18

Dr. Heinz Doofenshmirtz

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Multiply the "numerator" by $4$. – Daniel Fischer May 15 '20 at 18:20
@DanielFisher Why can I do that and why does that bring me furhter? – Dr. Heinz Doofenshmirtz May 15 '20 at 18:51
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You can do that because $\bigl(\frac{4}{m}\bigr) = 1$ for every odd $m$ and the Jacobi symbol is multiplicative. If you do that (right at the start) and look closely, you will see a substantial simplification. – Daniel Fischer May 15 '20 at 18:53
Not quite, the "numerator" becomes $4n^4 + 4n^2 + 4 = (2n^2+1)^2 + 3$. But $\bigl(\frac{3}{2n^2+1}\bigr)$ is much easier to evaluate (and it will be zero if $3 \nmid n$). – Daniel Fischer May 15 '20 at 19:18
I see now. Thank you! :-) – Dr. Heinz Doofenshmirtz May 15 '20 at 19:20

Calculate Jacobi symbol

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