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### Infinite Primes

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Date: Sun, 4 Dec 1994 19:40:57 -0800 (PST)
From: "Matthew K."
Subject: Largest Prime.

Hello,

I am a middle school math teacher. I have been sharing the history
of the search for the largest prime # with my students. Unfortunately, the
latest info I have is from 1979 (The Joy of Mathematics - T. Pappas). 2
to the power 44497 - 1.

We would appreciate it if you could send us the largest prime to date, and
how many digits it has.

In a related question, I would like to know how they compute primes if it
is not too involved. Several students want to know how they find
them.

Thank you, Matt Storms

Maze Middle School
Hollister, CA
```

```
Date: 5 Dec 1994
From: Dr. Math
Organization: Swarthmore College
Subject: Re: Largest Prime.

Hello there!

I just wanted to let you know that we received your message, and that
we'll get back to you very soon.  I do happen to have the largest known
prime written down in a book, but unfortunately that book isn't with me
right now.  I'll fetch it soon, though, and I'll let you know what I
find.  From what I remember, and from what other people around me
have said, it has about 200,000 digits or so.

It is important to remember that there really isn't a largest prime; there
are an infinite number of primes, and that's a proven fact.  But obviously
we don't know what all of them are, we only know how to show that
some special big numbers are prime.

Anyway, you'll be hearing from us again soon!
```

```
Date: Mon, 5 Dec 1994 19:53:57 -0800 (PST)
From: "Matthew K."
Subject: Re: Largest Prime.

Hello,

Thank you for the quick response. I was unaware of a proof for
the existence of an infinite number of primes. I would like to know the
gist of the proof, if possible. My students will want to know about it
when I tell them it's been proven there is an infinite number of primes. I
have a degree in math & have become pretty good at paraphrasing this
sort of thing for my students.

Thank you again.

Matt Storms

Maze Middle School
Hollister, CA
```

```
Date: Tue, 6 Dec 1994 01:40:49 -0500 (EST)
From: Dr. Ken
Subject: Re: Largest Prime.

Hello there!

Sure, I'll give you a proof that there are an infinite number of primes.

Let's suppose that there are only n primes (for classroom purposes,
you could just let n = 7 or something).  We'll see that this leads to a
contradiction.  Call the primes p1, p2, p3, ..., pn.  Then consider the
number (p1 x p2 x p3 x p4 x ... x pn) + 1.  It's not divisible by any of the
primes we have, since it's one more than a multiple of p1, one more than a
multiple of p2, and so on.  So this new number is either prime, or it's
divisible by some prime other than those n primes that we know about.  In
either case, we've found a new prime that wasn't in our list.  But this is a
contradiction, since those were ALL the primes.  Therefore, our initial
assumption must have been wrong, and there must be an infinite number of
primes.

I hope that's clear enough for you; it's kind of late, and I'm a little
tired, so please write back if you need more explanation.

-Ken "Dr." Math
```

```
Date: Tue, 13 Dec 1994 19:01:28 -0800 (PST)
From: "Matthew K."
Subject: Largest Prime

Hello,

I inquired last week as the to largest prime found to date. I have
not recieved this number yet, so I am asking again for the largest
prime & how many digits it has.

My 7th & 8th grade classes are very interested in this number & I
have not been able to find it locally.

Thank you, Matt Storms

Maze Middle School
```

```
Date: Wed, 14 Dec 1994 15:42:19 -0500 (EST)
From: Dr. Ken
Subject: Re: Largest Prime

Hello there!

I'm sorry I didn't get back to you sooner.  The newsreading program
we use to address people's questions somehow went berserk and deleted

But never fear!  Here's the largest number that we know for certain is
prime:  2^216091 - 1.  That's 1 less than 2 to the 216091 power.  By my
calculations (you take the log base ten of this number, or rather this
number plus one, since they have the same number of digits), it has
65050 digits.

Thanks for writing, and please write back if you have more questions!

-Ken "Dr." Math
```
Associated Topics:
Middle School Prime Numbers

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