14.4. 29 ** How many prime numbers have a decimal representation that is exactly alternating 0s and 1s, and starts and ends with 1?
Let $p_{k}=1010 \cdots 101(k$ zeros, $k \geqslant 1)$. Clearly, $p_{1}=101$ is a prime number. If $k$ is odd, then $101 \mid p_{k}$. If $k$ is even, then $11 p_{k}=\underbrace{11 \cdots 11}_{2+2 \uparrow 1}=\underbrace{11 \cdots 1}_{k+1 \uparrow 1} \times \underbrace{100 \cdots 01}_{k \uparrow 0}$. And $11 \mid 100 \cdots 01$, so $p_{k}=\underbrace{11 \cdots 1}_{k+1 \uparrow 1} \times \frac{100 \cdots 01}{11}$ is a composite number. Therefore, only the prime number 101 satisfies the requirement.