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The main question is : Is $545^4 + 4^{545}$ a prime number? Explain your answer.

My approach : I tried writing the expression as, $$545^4 + 4*4^{544}$$ Thus we get, $$545^4 + 4*{(4^{136})}^4$$

I can't proceed any further. Is there some obvious thing or concept I'm missing? This question is a base-level olympiad question, thus there must be some short solution to this. I will appreciate if you can give a detailed answer, along with a shortcut if you have one, so that I understand the concept thoroughly. Thanks!

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2 Answers

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We can go through this using primitive steps. (Proceeding after the two steps given in my approach) Let $545=x$ and $4^{136}=y$. Thus, the expression becomes : $$x^4+4y^4$$ Adding and subtracting by $4x^2y^2$, we get, $${(x^2+2y^2)}^2-{(2xy)}^2$$ This can be written as : $$(x^2+2y^2+2xy)(x^2+2y^2-2xy)$$

Now, since there are two factors to the original expression excluding $1$ and itself, the expression $545^4+4^{545}$ is not a prime.

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$$ x^2 - 2 x y + 2 y^2 = (x-y)^2 + y^2 \geq y^2 $$

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