<p>How do you prove that there is no largest prime?</p>
<p>I have a kind of solution to this but it isn’t really very solid. Could anyone help please?</p>
<p>Assume that there is such number as the largest prime number, and let P be it.
Let Q = product of all primes, up to and including P, +1 (i.e. 1x3x5x7x11x…xP+1)
Q cannot be divided by any prime number smaller than it without leaving a remainder of 1.
Q, therefore, is prime, and is also larger than P, which contradicts our choice of P as the largest prime.
Therefore no such number exists.</p>
<p>interesting</p>
<p>Theoretically that proof should work for all numbers, right? Because, for instance, 1 times 3 timese 5 plus 1 = 16 - divisible by 4 and 2. I don’t get how you know that the product plus one isn’t divisible by any non-primes (which is why i did an easy example and then I got confused)</p>
<p>Cevonia:</p>
<p>Your proof is right, except the “(i.e. 1x3x5x7x11x…xP+1)” part. “1” is not a prime, and you missed the “2.” So here’s the modified proof:</p>
<p>Suppose a largest prime P exists. Let Q=1+(product of all primes up to P). Q has a remainder of 1 when divided by any of the primes, so Q must by a prime. But we just said P is the largest prime, and Q>P, so this is a contradiction. Therefore a largest prime doesn’t exist.</p>
<p>Prettyfish:</p>
<p>If a number Q is not divisible by any member of the set S, then it can’t be divisible by any product of members in S. For example, if Q isn’t divisible by 2,3,or 5, then it’s not going to be divisible by 2*3. So you don’t need to even consider composites.</p>
<p>mrang space</p>
<p>tsunashima1, that’s what I thought at first, too. That thing about if you multiply all these numbers together and add 1, that number should have a remainder of 1 when divided by the original factors. However, for some reason, it doesn’t work with the number 2, which is a prime number. </p>
<p>But, suppose you had the largest prime. That prime number would have to be an odd number, P. If you say a large prime is Q = P + 1, then Q would be even and divisible by 2. Therefore, Q would not be prime. </p>
<p>However, this doesn’t not prove the non-existence of the largest prime number; it just shows that the above method is not an effective one in solving the solution. </p>
<p>By the way, I think that this question is still something yet unsolved by mathematicians, so if you could find the solution, it would be a pretty big deal. It’s not something that will come up on a (normal) math test.</p>
<p>Suburbian,</p>
<p>Mathematicians have actually solved this problem, and the proof I provided is the official one.</p>
<p>“However, for some reason, it doesn’t work with the number 2, which is a prime number.”</p>
<p>Don’t forget that the proof specifies Q as 1+(product of primes). The number “2” doesn’t fit that description.</p>
<p>"But, suppose you had the largest prime. That prime number would have to be an odd number, P. If you say a large prime is Q = P + 1, then Q would be even and divisible by 2. Therefore, Q would not be prime. "</p>
<p>I never said Q = P+1. I said Q = 1+(product of all primes up to P).</p>
<p>Oh yeah, by the way I actually did receive this problem on my math test two years ago. My teacher was an International Math Olympiad gold medalist :)</p>
<p>Ok, that makes more sense. It’s finals week and thus my brain doesn’t quite function properly and I completely forgot about 1 and 2.</p>
<p>Whoops. Brain was tired. Missed number 1, meant 2. Sorry.</p>
<p>thanks people!</p>
<p>Whoops, totally missed the “product” part! That would change everything.
But yeah, last year at school, I saw a video about the world’s greatest mathematical puzzles, and this problem was presented as one of them! I assumed the video was made recently, but I guess not. And someone else was actually talking to me about this prime number question this spring (not at school), and I remember thinking that it made sense. Then, when I (mis)read these posts and after I made my post, I was thinking how what the other person said to me and what tsunashima and some others said seemed really similar and why I didn’t catch the mistake in the spring… anyway, I’m babbling. I tend to do that!</p>
<p>Also, response to “My teacher was an International Math Olympiad gold medalist”: haha, my teacher was a self-described average student in high school and often likes to say that he doesn’t want to be “too intellectual.”</p>
<p>“But yeah, last year at school, I saw a video about the world’s greatest mathematical puzzles, and this problem was presented as one of them!”</p>
<p>LOL. Euclid wrote the proof in 300 BC.</p>
<p>It probably wasn’t the world’s greatest /unsolved/ mathematical problems?</p>
<p>But usually the term “greatest mathematical puzzles” refers to math problems that are either unsolved (Twin Prime Conjecture) or have been solved only recently (Fermat’s Last Theorem).</p>
<p>Hmm. Maybe the makers were just smoking crack?</p>
<p>That’s certainly a possibility :)</p>
<p>Hm. It’s been eleven years since FLT was solved… it doesn’t seem that long</p>
<p>oh, no it was a video about UNsolved puzzles.</p>
<p>but I admit, I watched it in a theory of knowledge class, not a math class, and my theory of knowledge teacher is actually a history teacher…</p>