MHB Proving $(a+b,\ a^2-ab+b^2)=1$ or $3$

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If \( (a, b) = 1 \), then \( (a+b, a^2-ab+b^2) \) must equal either 1 or 3. The expression simplifies to \( (a+b, 3ab) \), leading to two cases based on the divisibility of \( a+b \) by 3. If \( (a+b, 3) = 1 \), then \( (a+b, 3ab) = 1 \) follows. Conversely, if \( (a+b, 3) = 3 \), it implies that \( 3 \) divides \( a+b \), confirming the result. The proof effectively shows that the greatest common divisor can only be 1 or 3 under the given conditions.
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Show that if $(a, b)=1$ then $(a+b,\ a^2-ab+b^2)=1\text{ or }3$.

My attempt:

$(a+b,\ a^2-ab+b^2)=(a+b,\ a^2-ab+b^2-(a+b)(a+b))=(a+b,\ -3ab)=(a+b,\ 3ab)$

$(a,\ a+b)=(b,\ a+b)=1$

$\therefore (a+b,\ ab)=1$

Consider 2 cases:

1) $(a+b,\ 3)=1 \implies(a+b,\ 3ab)=1$

2) $(a+b,\ 3)=3\implies 3|a+b$

$\displaystyle (a+b,\ ab)=1\implies\left(\frac{a+b}{3},\ ab\right)=1$

$\displaystyle (a+b,\ 3ab)=3\left(\frac{a+b}{3},\ ab\right)=3$

Is that ok?
 
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Alexmahone said:
Show that if $(a, b)=1$ then $(a+b,\ a^2-ab+b^2)=1\text{ or }3$.

My attempt:

$(a+b,\ a^2-ab+b^2)=(a+b,\ a^2-ab+b^2-(a+b)(a+b))=(a+b,\ -3ab)=(a+b,\ 3ab)$

$(a,\ a+b)=(b,\ a+b)=1$

$\therefore (a+b,\ ab)=1$

Consider 2 cases:

1) $(a+b,\ 3)=1 \implies(a+b,\ 3ab)=1$

2) $(a+b,\ 3)=3\implies 3|a+b$

$\displaystyle (a+b,\ ab)=1\implies\left(\frac{a+b}{3},\ ab\right)=1$

$\displaystyle (a+b,\ 3ab)=3\left(\frac{a+b}{3},\ ab\right)=3$

Is that ok?
Yes. Perfect.
 
You might be interested in the following approach:

Let $d$ be a common divisor of $a+b$ and $a^2-ab+b^2$. Then $ d|(a+b)$ and this can be rewritten as $ a \equiv - b (\bmod. d)$

And so $0\equiv a^2-ab+b^2 \equiv (-b)^2 - (-b) \cdot b + b^2 = 3 b^2 (\bmod. d)$ . But we must have $(d,b) = 1$ since $(a,b)=1$ and $d | (a+b)$ (because any common divisor of $b$ and $d$ will also divide $a$).

Thus $0 \equiv 3 (\bmod. d)$ ...
 
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