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My question about primes

Are there infinite primes? If yes, or no, could you prove it? Any useful internet resources?

Is there a usual pattern for primes?

How to define Mersenne primes? ( \( M_n=2^n-1) \). And how many of them had been found?

And others? Like Wilson's Theorem \( (n-1)! \equiv -1 (\text{mod n}) \)? Or all coefficients of \( (x-1)^y -(y^p)-1 \) is divisible by \( y \), then \( y \) must be prime.

Recently, I knew that the largest known prime number, which is discovered in August \( 2015 \) by Great Internet Mersenne Prime Search, is \[ \displaystyle \Huge{ \color{green}{2}^{\color{blue}{57,885,161}} - \color{red}{1}} \]It contains \( 17, 425,170 \) digits! It's a Mersene prime.

Also, the largest non-Mersenne prime number is:

\[ \Huge \color{green}{19249} \color{red}{\times 2^{\color{blue}{13018586}}} + 1 \]

Note by Adam Phúc Nguyễn
1 year, 2 months ago

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I was excited to hear they had discovered a new larger Mersenne prime. Then I read closer and found that this is the largest known prime so far as of this month.

It was actually discovered back in 2013. Seems like it's past time they found a bigger one, right? Steven Perkins · 1 year, 2 months ago

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Nope, there is no pattern to primes, in the sense that there's no simple formula that we can plug numbers into to calculate the nth prime.

However, there are some cool patterns relating to the distribution of primes among the integers. I vaguely remember a nice YouTube video about that: I'll link it here if I can find it... Raj Magesh · 1 year, 2 months ago

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Yes there are infinite primes. There are some proofs in the brilliant wikis and obviously on the wikipedia too. Here is a link-- Infinitely Many Primes Ashwin Upadhyay · 1 year, 2 months ago

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@Ashwin Upadhyay FYI, you can use our nifty wiki linking tool to easily link to wiki pages

Calvin Lin Staff · 1 year, 2 months ago

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@Calvin Lin Infinitely Many Primes

understood. Thanks. Ashwin Upadhyay · 1 year, 2 months ago

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