Let's assume there is a prime which can be expressed \(a^4 - b^4\). Since prime numbers are greater than 1, then \(a^4 \gt b^4\) and \(a\) cannot be 1 or 0. We can also express \(a^4 - b^4\) as:
$$(a^2 - b^2)(a^2 + b^2)$$
If \(a^4 - b^4\) then one of the factors is a prime and the other is 1. Since \(a\) cannot be 1 or 0, then \(a^2 \gt 1\), so \(a^2 + b^2\) can't be 1, which means:
$$ (a^2 - b^2) = 1 $$$$ (a + b)(a - b) = 1 $$$$ (a + b) = (a - b) = ±1 $$
The above statement is only possible if \(a=±1\) and \(b=0\), but this cannot be true, otherwise \(a^4 - b^4\) would be 1 (a non-prime). This is a contradiction. So no primes can be expressed as \(a^4 - b^4\).