May 2025
There is a solution to \(m^2 - 8e^2 = 1\) where both \(m\) and \(e\) are integers between \(0\) and \(5\). Can you somehow use your solution for the equation \(m^2 - 8e^2 = 1\) to find solutions to the equations \(m^2 - 8e^2 = 4\) and \(m^2 - 8e^2 = 9\)?
There is a pair of integers \((m,e)\) satisfying \(m^2 - 8e^2 = 49\) where both \(m\) and \(e\) are single-digit positive integers.
Try expanding out \(N\big((a + b\sqrt 8)(c + d\sqrt8)\big)\) and \(N(a + b\sqrt 8)N(c + d\sqrt 8)\).
Expand out \(N(a + b\sqrt 8)\) and use the result from Question 3.
You will need to apply Question 3 repeatedly here. Consider your \(m\) and \(e\) from Question 2. What is \(N((m+e\sqrt 8)^2)\)? What about \(N((m + e\sqrt 8)^{2^n})\)?