Problem #1

[RMO-1991] : Prove that $$n^4 + 4^n$$ is composite for all integer values of n greater than 1.

Solution:

The two basic ways of proving that an expression is composite are:

1. Prove that a prime number always divides the given expression
2. The expression can be factored into multiple factors.

The second approach works in this case. The identity $$(a^2 - b^2) = (a - b)(a + b)$$ is the often useful in such cases. A natural way to proceed is to try to convert the given expression into a format closer to $$(a^2 - b^2)$$

That’s pretty much it! Note that we used the fact that $$n$$ is odd; this made $$n^2.2^{n+1}$$ a perfect square. When $$n$$ is even, the given expression is also even and hence composite.