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# It's Prime Time!

I found these interesting facts from a Number Theory book. Can you prove it? 1) Prove that if $$\dfrac{3^k + 1}{2}$$ is a prime, then $$k$$ is a power of $$2$$. 2) Prove that if $$\dfrac{3^k - 1}{2}$$ is a prime, then $$k$$ is a prime.

Note by Marc Vince Casimiro
3 years, 3 months ago

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### 1

Let $$k=2^x \times y$$, where $$y$$ is odd. If, on contrary, $$k$$ weren't a power of $$2$$, $$y$$'d be an odd number greater than $$1$$. Then $${3}^{k+1}={3}^{2^x.y}+1=({3}^{2^{a}})^{y}+{1}^{y}$$ would be divisible by $${3}^{{2}^{a}}+1$$ , because $$y$$ is odd, and would be divisible by $$2$$, which cancels the $$2$$ in the denominator, and another factor. One more factor would be in the second factor in our factorization, thus making it composite. Hence, our assumption is wrong.

### 2

Similar to 1.

- 3 years, 3 months ago