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 $\displaystyle \Huge{ \color{#20A900}{2}^{\color{#3D99F6}{57,885,161}} - \color{#D61F06}{1}}$It contains $17, 425,170$ digits! It's a Mersene prime.

Also, the largest non-Mersenne prime number is:

$\Huge \color{#20A900}{19249} \color{#D61F06}{\times 2^{\color{#3D99F6}{13018586}}} + 1$ 5 years, 11 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?

- 5 years, 11 months ago

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

- 5 years, 11 months ago

FYI, you can use our nifty wiki linking tool to easily link to wiki pages Staff - 5 years, 11 months ago

understood. Thanks.

- 5 years, 11 months ago

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...

- 5 years, 11 months ago