Max and Bob!

Since 2 is the only even prime and it's obviously less than 100, therefore we are searching for an odd perfect square containing the digit 4 that's less than 100. The answer is 7² = 49 < 100.

Let $x$ be Max's favorite number.

If questions 2 and 4 are both answered truthfully then $x=2$ . But, if those are both true answers, 1 must be answered falsely and $x\geq 100$ , a contradiction.

Thus, we know that Max's answers to questions 1, 3, and 5 are true and his answers to 2 and 4 are false. So $x<100$ , $x$ is not prime, $x$ contains the digit four, $x$ is not even, and $x$ is a perfect square.

The only odd square less than one hundred that contains the digit 4 is $\boxed{49}$ .

First, let us assume that Max first tells the truth. This means that the number is less than 100, but it is prime and even. The only even prime is 2, and it is less than 100. This is a contradiction, so the first statement must be a truth, and the number is less than 100. The only perfect squares under 100 than contain the digit 4 are 4, 49, and 64. However, Statement 4, which is a lie because Statement 1 is a truth, says that the number is even, which means that the number is actually odd. So Max's number is 49.