Infinite PrimesDate: 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. If I'm not mistaken, the old Greeks knew about this one. 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 your message. 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 |
Search the Dr. Math Library: |
[Privacy Policy] [Terms of Use]
Ask Dr. Math^{TM}
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/