급수와 소수

$\displaystyle \sum\limits _{n=1} ^{\infty} \left( \frac{a}{6 ^{2n-1}} + \frac{b}{6 ^{2n}} \right)$ $\displaystyle = \left( \frac{a}{6} + \frac{a}{6 ^{3}} + \frac{a}{6 ^{5}} + \cdots \right)$$\displaystyle + \left( \frac{b}{6 ^{2}} + \frac{b}{6 ^{4}} + \frac{b}{6 ^{6}} + \cdots \right)$ $\displaystyle = \frac{\frac{a}{6}}{1 - \frac{1}{6 ^{2}}} + \frac{\frac{b}{6 ^{2}}}{1 - \frac{1}{6 ^{2}}}$ $\displaystyle = \frac{6 a + b}{6 ^{2} - 1}$ $\displaystyle = \frac{6 a + b}{35}$$\displaystyle = \frac{1}{p}$ ∴ $\displaystyle p = \frac{35}{6 a + b}$$\displaystyle = \frac{5 \times 7}{6 a + b}$ 따라서, $p$는 소수이므로 $6 a + b = 7$일 때 $p = 5$이고, $6 a + b = 5$일 때 $p = 7$이다. ∴ $5 + 7 = 12$