수학문제집 풀이

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$M{M}_p=2^{2^p-1}-1$

$M{M}_{M_p}=2^{2^{2^p-1}-1}-1$

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$\left(\frac{3}{M{M}_p}\right)=-1$ 일 때 $\left(\frac{3}{M{M}_{M_p}}\right)=-1$ 임을 보여라.

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**Proof**:

$M{M}_p=2^{2^p-1}-1$

$M{M}_{M_p}=2^{M{M}_p}-1$

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$\left(\frac{3}{M{M}_p}\right)=-1$ 이므로 만족하는 $M{M}_p$를 찾는다. 이 때, $x=12k\pm 5$ (k는 정수)

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$2^x-1=2^{12k\pm 5}-1$

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$2^{12k+5}-1=1+2^1+2^2+...+2^{12k+4}$

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$\equiv 1+2+4+8+...+8+4(\mathrm{mod}12)$

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$\equiv 7(\mathrm{mod}12)$

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따라서, 

$\left(\frac{3}{2^x-1}\right)=-1$

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이렇게 정리할 수 있습니다.