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Problem #1

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


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.