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.

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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. – Satvik Golechha · 2 years, 1 month ago

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