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Checking on some convergent series

I’m self-studying Ireland/Rosen, but question #5.36 doesn’t make sense to me. It asks

Show that part (c) of Proposition 5.2.2 is true if $a$ is negative and $b$ is positive (both still odd).

Part (c) of the Proposition says if $a$ is odd and positive as well as $b$, then

$$

\left(\frac{a}{b}\right)\left(\frac{b}{a}\right)=(-1)^{((a-1)/2)((b-1)/2)}.

$$

By the way, these are Jacobi symbols, not Legendre symbols. But if $a$ is negative, then $\left(\frac{b}{a}\right)$ isn’t defined, as far as I know, so I don’t understand what I’m supposed to prove. Am I misunderstanding the question? Thanks.

- Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
- Prove: The product of any three consecutive integers is divisible by $6$.
- Prove that $x^4+x^3+x^2+x+1 \mid x^{4n}+x^{3n}+x^{2n}+x^n+1$
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- Is knowing the Sum and Product of k different natural numbers enough to find them?
- Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction

There is the Kronecker symbol.

A different approach is to define $\left( \frac{a}{-p} \right)$ the same way you do $\left( \frac{a}{p} \right)$; after all, $-p$ is a prime number, and “modulo $-p$” works out to the same thing as “modulo $p$”.

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