Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?
To determine if there exists a positive integer \( m \) such that \( S_{m+1} = 4S_m \), where \( S_n \) denotes the least common multiple (LCM) of \( 1, 2, \ldots, n \), we will analyze the properties of the LCM function and the given condition.

1. **Express \( S_n \) in terms of prime factors:**
   \[
   S_n = \text{lcm}(1, 2, \ldots, n)
   \]
   The LCM of the first \( n \) natural numbers is the product of the highest powers of all primes less than or equal to \( n \).

2. **Consider the condition \( S_{m+1} = 4S_m \):**
   \[
   S_{m+1} = \text{lcm}(1, 2, \ldots, m, m+1)
   \]
   For \( S_{m+1} \) to be exactly four times \( S_m \), the number \( m+1 \) must introduce a factor of 4 that is not already present in \( S_m \).

3. **Analyze the prime factorization of \( m+1 \):**
   Let \( m+1 = p_1^{z_1} p_2^{z_2} \ldots p_k^{z_k} \), where \( p_i \) are distinct primes and \( z_i \) are positive integers. We need to ensure that \( S_{m+1} = 4S_m \).

4. **Consider the prime factor 2:**
   For \( S_{m+1} = 4S_m \), the factor of 4 must come from the prime factor 2. This means \( m+1 \) must be of the form \( 2^a \) where \( a \geq 2 \).

5. **Check the highest power of 2 in \( S_m \):**
   Let \( v_2(n) \) denote the highest power of 2 dividing \( n \). For \( S_m \), the highest power of 2 is \( 2^{\lfloor \log_2 m \rfloor} \).

6. **Determine the condition for \( S_{m+1} = 4S_m \):**
   \[
   S_{m+1} = \text{lcm}(S_m, m+1)
   \]
   For \( S_{m+1} = 4S_m \), \( m+1 \) must introduce exactly two additional factors of 2. This implies:
   \[
   m+1 = 2^{\lfloor \log_2 m \rfloor + 2}
   \]

7. **Check if such \( m \) exists:**
   \[
   m+1 = 2^{\lfloor \log_2 m \rfloor + 2}
   \]
   This equation implies that \( m \) must be of the form \( 2^k - 1 \) for some integer \( k \). However, substituting \( m = 2^k - 1 \) into the equation \( m+1 = 2^{\lfloor \log_2 m \rfloor + 2} \) does not yield a consistent solution.

8. **Conclusion:**
   Since no such \( m \) satisfies the condition \( S_{m+1} = 4S_m \), we conclude that no such positive integer \( m \) exists.

\[
\boxed{\text{None}}
\]