# primes proofs observations [closed]

1) Taking $p_1,p_2,\dots,p_k$ to be the primes up to $x^{1/2}$ we have a way to determine, with proof, each prime $N$ between $x^{1/2}$ and $x$ by Finding a representation of $N$ as in part c. Find all the primes between $5$ and $100$ in this way, along with these proofs that they are indeed prime.

2) Use the sequence of Fermat numbers to prove that for each integer $k \ge 1$, there are infinitely many primes are congruent to $1 \pmod {2^k}$.

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## closed as off-topic by Thursday, Ivo Terek, Jonas Meyer, Jack D'Aurizio, J. W. PerryAug 11 at 4:08

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• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Thursday, Ivo Terek, J. W. Perry
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The usual question: what have you attempted? –  Ｊ. Ｍ. Oct 6 '11 at 8:02
@JM! I want to say in different ways that primes are infinite. I just failed to prove it. –  gandhi Oct 6 '11 at 8:05
What is "part c"? –  Gerry Myerson Oct 6 '11 at 11:46

I assume this problem comes from, or is based on, some book. Is there something in the book about factors of $2^{2^k}+1$ all being congruent to 1 modulo some power of 2? Is there something in the book about every number of the form $2^{2^k}+1$ being relatively prime to every other number of that form? And can you put these two things together to get what you want?