Let $n, m$ be positive integers such that
\[n(4n+1)=m(5m+1)\]
(a) Show that the difference $n-m$ is a perfect square of a positive integer.
(b) Find a pair of positive integers $(n, m)$ which satisfies the above relation.	

Additional part (not asked in the TST): Find all such pairs $(n,m)$.
Given the equation:
\[ n(4n + 1) = m(5m + 1) \]

### Part (a): Show that the difference \( n - m \) is a perfect square of a positive integer.

1. **Assume \( m > n \)**:
   Let \( d = \gcd(m, n) \) and write \( m = dm_1 \) and \( n = dn_1 \). Substituting these into the given equation, we get:
   \[
   d n_1 (4d n_1 + 1) = d m_1 (5d m_1 + 1)
   \]
   Dividing both sides by \( d \), we obtain:
   \[
   n_1 (4d n_1 + 1) = m_1 (5d m_1 + 1)
   \]

2. **Rearrange the equation**:
   \[
   4d n_1^2 + n_1 = 5d m_1^2 + m_1
   \]
   \[
   n_1 - m_1 = d (5m_1^2 - 4n_1^2)
   \]

3. **Consider the prime divisors**:
   Let \( p \) be a prime divisor of \( m_1 - n_1 \). Then:
   \[
   m_1 \equiv n_1 \pmod{p} \implies 5m_1^2 - 4n_1^2 \equiv n_1^2 \not\equiv 0 \pmod{p}
   \]
   This implies \( 5m_1^2 - 4n_1^2 = 1 \).

4. **Solve the equation**:
   \[
   5m_1^2 - 4n_1^2 = 1
   \]
   This is a form of the negative Pell's equation. The general solution to this equation can be found by solving:
   \[
   (2n_1)^2 - 5m_1^2 = -1
   \]

5. **Find the relationship between \( n \) and \( m \)**:
   From the equation \( 5m_1^2 - 4n_1^2 = 1 \), we have:
   \[
   d = n_1 - m_1
   \]
   Therefore:
   \[
   n - m = d^2
   \]
   Hence, \( n - m \) is a perfect square.

### Part (b): Find a pair of positive integers \( (n, m) \) which satisfies the above relation.

1. **Solve the Pell's equation**:
   The negative Pell's equation \( (2n_1)^2 - 5m_1^2 = -1 \) has a known solution:
   \[
   (n_1, m_1) = (19, 17)
   \]

2. **Calculate \( n \) and \( m \)**:
   Using \( d = n_1 - m_1 \):
   \[
   d = 19 - 17 = 2
   \]
   Therefore:
   \[
   n = d n_1 = 2 \times 19 = 38
   \]
   \[
   m = d m_1 = 2 \times 17 = 34
   \]

The final answer is \( \boxed{ (n, m) = (38, 34) } \)