Question

\( p \) is a prime and \( m \) is a positive integer. How many solutions exist for the equation \[ p^5 - p = (m^2 + m + 6)(p - 1)? \]

• A. 0
• B. 1
• C. 2
• D. Infinite

Hint:

Try small values of \( p \) and check whether \( m^2 + m + 6 \) becomes integer.

Answer 1

The Correct Option is B

We solve: \[ p^5 - p = (m^2 + m + 6)(p - 1) \Rightarrow \frac{p^5 - p}{p - 1} = m^2 + m + 6 \] Try small primes: For \( p = 2 \): \( \frac{32 - 2}{1} = 30 \) → \( m^2 + m + 6 = 30 \Rightarrow m^2 + m - 24 = 0 \Rightarrow m = 4 \) (valid)
For larger primes, LHS grows too rapidly. So only 1 valid \( m \).
\[ \boxed{1 \text{ solution}} \]