Mathematical Proof ...that there is no largest prime number

[RedAlert 0]

In math, a proof is basically just that, proof that something is true. Proofs must be rigid, they cannot be based on assumptions. For example, just because we keep finding bigger and bigger prime numbers, it does not mean there are infinitely many. However, there is a proof for that.First, assume that there is, in fact, a largest prime number. Call it "P." That is, there is no prime that is greater than P. (It's OK to assume things like this in a proof sometimes; for example, it's OK here because we intend to prove that our assumption is wrong, that it leads to an impossible situation.)Then, consider the product of all prime numbers up to P. That is, 2x3x5x7x11x13x...xP. Now add 1 to that product.The result is a number that, if divided by any prime number, yields a remainder of 1, and therefore is itself prime.BUT WAIT! P is supposed to be the largest prime number! But we just found one bigger than P! And that impossible situation is exactly what proves that there is in fact no largest prime number.*NOTE*For some prime P, the number 2x3x5x7x11x13x...xP + 1 is not necessarily itself prime. There can be more primes between that number and P which are factors of 2x3x5x7x11x13x...xP + 1.

[matak 2]

I can't belive that there i no answer to this post.The assumption "There is no largest prime number!". OK, but from what number.. :unsure:Well, i find this proof really funny and i just had to reply to it. I just don't know the real reason to answer it but other than this is something really funny, and i hope that more people will read it.

[hitmanblood 0]

OK I see the need to clarify certain things here. As firstly you just didn't make proof of the biggest number and some other things. More or less you jsut made mess here.Mathematical proof is results of deductive reasoning from assumtions or axioms as they are more or less usually called in mathematics. Axioms are assumptions which haven't been and cannot been prooved to be right or wrong we just begin with that fact that they are right. In the XX century there is hall new mathematics which works with axioms and changes them and then check what they get and so on. Then again the thing you are talking in your so called proof is actually redution ad absurdo or in english reduction to the absurd. You deliberately choose wrong premise and then work your way to prove that this premise is wrong and by doing so proving that in fact your initial premise or the thing you wanted to prove is correct. You can use this wherever you like in math but it will not be successful everywhere. And then to tell you whats wrong in your proof it is that you have multiplied prime numbers let's take that number 29 is prime number (it is actually) and that the biggest prime number that is 29 is biggest prime number this is our premise. And by your premise we should obtain number 30. Then we calculate in the way you explained:1*2*3*5*7*11*13*17*19*23*29+1 = 6469693231 hopefully I calculated all this correctlyAnd now I will explain what prime numbers are. Prime numbers are all those numbers that are divided only with 1 and the number itselfOK so this result as you said is one larger then 29 so the result should be 30 and it is really larger number then number 30. But let's leave this aside and check the other important thing is a prime number that is if it is only divisible by 1 and itself. So I wrote small program long time ago and put this number in it and guess what it failed. What a suprise it is not prime number.6469693231 is divisible by 3. Which proves that your prof is wrong.WRONG

[RedAlert 0]

I know that this thread is old, but I just wanted to point out that 6469693231 is not divisible by 3, and anyone who knows how to divide should know that.

[suberatu 0]

I am offering $5 to whoever can prove Fermat's Last Theorem without making any references to modular functions or elliptical curves.

[pop 0]

I am offering $5 to whoever can prove Fermat's Last Theorem without making any references to modular functions or elliptical curves.

can you be a bit more specific on that subject. how about if you try explain what is that you need to be proved, couse i read it already proved but looks similar to fake.

[suberatu 0]

can you be a bit more specific on that subject. how about if you try explain what is that you need to be proved, couse i read it already proved but looks similar to fake.

Sorry, I guess I really should have been more specific. Basically, Fermat's Last Theorem states that aⁿ + bⁿ = cⁿ is not true for any integer n greater than 0. That is, a² + ² = c² is true, but if you change the power to any integer greater than 2, you will never get values for a, b, and c that make the equation true (the only exception is when you get a rounding error from a basic handheld calculator). This was a problem that had previously gone unsolved for centuries, until Andrew Wiles proposed a proof (which he re-edited a year or two after the original publication) to prove the theorem. Two of the main elements of his proof were modular functions and elliptical curves. I'm not a mathematical prodigy, so I can't really explain what they are very accurately, but there is a 5 part documentary on YouTube which explains the theorem and how Wiles came to solve it (Link to Part 1). I highly recommend that anyone with the slightest interest in mathematics watch this documentary. It is very educational and interesting as well.

 

P.S.: I have never actually seen the proof with my own eyes, but it is said that it is around the area of 200 pages, and that its mathematics are so complex that the majority of people living on earth would not understand what they are reading. Seeing how complex and long the actual proof is (which took Wiles over 7 years to formulate), I was actually joking about the $5 offer for another proof, since it is nearly impossible, and anyone who can actually formulate another is bound for much greater things than a $5 bill.

[salamangkero 0]

OK I see the need to clarify certain things here. As firstly you just didn't make proof of the biggest number and some other things. More or less you just made mess here.

I'd like to point out that this is a bit too hostile...

 

Then again the thing you are talking in your so called proof is actually redution ad absurdo or in english reduction to the absurd. You deliberately choose wrong premise and then work your way to prove that this premise is wrong and by doing so proving that in fact your initial premise or the thing you wanted to prove is correct. You can use this wherever you like in math but it will not be successful everywhere.

What you just said was irrelevant. First, I'd like to clarify what RedAlert was trying to prove "There is no such thing as the largest prime number." Hope we're clear on that?

 

Next, his premises were outlined as "If we multiply all prime numbers up to P, and add 1 to the product, the resulting number would be prime," which, as far as I know, is correct.

 

If you meant reduction ad absurdum, I think you are also aware that it is, for all intents and purposes, a valid way to prove something. It is not, repeat, is not deliberately choosing a wrong premise and by proving it wrong, effectively supporting the veracity of the opposite premise. Au contraire, it is proving some theorem right by assuming the opposite (false) outcome (repeat, outcome and not premise) and proving that, given the premise, it is impossible to arrive at that outcome.

 

So, what is the opposite outcome? "There is such a thing as a largest prime number P." Now we work on the premise to try and achieve that opposite outcome. "If we multiple all prime numbers up to P and add 1 to the product, the resulting number would be prime." Now, given that the premise was true, the resulting number would be greater than P and, undoubtedly, this reduces the argument "P is the largest prime number" to an absurdity. I hope you do understand how reductio ad absurdum works, as any sane person who actually studied logic should.

 

And then to tell you whats wrong in your proof it is that you have multiplied prime numbers let's take that number 29 is prime number (it is actually) and that the biggest prime number that is 29 is biggest prime number this is our premise. And by your premise we should obtain number 30. Then we calculate in the way you explained:

 

1*2*3*5*7*11*13*17*19*23*29+1 = 6469693231 hopefully I calculated all this correctly.

First off, one is not a prime number. Any high school student should already be aware of that.

 

And now I will explain what prime numbers are. Prime numbers are all those numbers that are divided only with 1 and the number itself.

 

OK so this result as you said is one larger then 29 so the result should be 30 and it is really larger number then number 30.

Here lies your mistake. You add 1 to the product, not the last factor. You should learn to read other people's posts well before screaming out to the world they're wrong.

 

Then, consider the product of all prime numbers up to P. That is, 2x3x5x7x11x13x...xP. Now add 1 to that product.

 

The result is a number that, if divided by any prime number, yields a remainder of 1, and therefore is itself prime.

See what I mean? Product, man, product! You add 1 to 6469693230 not 29, 'ayt?

 

So I wrote small program long time ago and put this number in it and guess what it failed. What a suprise it is not prime number.

 

6469693231 is divisible by 3. Which proves that your prof is wrong.

Dude, have you ever considered that there's something wrong with the small "program" you "wrote"? 'coz I've already tried dividing 6469693231 by 3 using the calculators of MS Windows and Casio and, surprise, surprise, it is so, most definitely, not divisible by 3.

 

WRONG

Yes, you are, m'friend
Edited October 14, 2007 by salamangkero (see edit history)

[csp4.0 1]

you're forgetting the one thing. Infinite. It is a unknown number, I know that infinite is both odd, even, prime, non-prime and all the others you can think of. Because Infinity+1=Infinity, and so does infinity!, and so on...So that proves it, infinity is a prime and so there is no end to the number of primes there are...

[bthaxor 0]

can i just say that when i first read your topic title, i thought you were joking.the answer to your question is quite simple: of course there isn'tdidn't it ever pop into your mind that there is NO 'largest' number? as csp pointed out, there is no limit to the number of prime numbers there are, so it would therefore be madness to try and find the largest prime number!

[omarsdali 0]

6469693231 is divisible by 3. Which proves that your prof is wrong.

6469693230 is divisible by 3 not 6469693231. 6469693230 is divisible by 3 as well as all these prime numbers 2 5 7 11 13 17 19 23 29. BUT what happens if you add one ? you get a new prime number that can only be divided by 1 and itself.

So I wrote small program long time ago and put this number in it and guess what it failed. What a suprise it is not prime number.

You put the wrong number in your program, you put in 6469693230 instead of 6469693231=6469693230 + 1.
Edited December 3, 2007 by omarsdali (see edit history)

[cangor 0]

you're forgetting the one thing. Infinite. It is a unknown number, I know that infinite is both odd, even, prime, non-prime and all the others you can think of. Because Infinity+1=Infinity, and so does infinity!, and so on...So that proves it, infinity is a prime and so there is no end to the number of primes there are...

Infinity isn't even a number. My math teacher hates it when kids say it is. It's a concept, and so you can't say that infinity is prime. If anything, it's absolutely unprime and divisible by any number.

This is kinda just a weird example of a mathematical proof though. Some of them are actually for interesting things, like finding the value of the sum of the first n numbers, or 1+2+3+4...+n = 5.(n^2+n). You can prove that with a mathematical proof. You can also do proofs to get area formulas and stuff people use in real life at work and stuff, because I'm pretty sure there are like two people in the world whose job depends on the fact that there are an infinite number of primes.

I'm not sure if this prime number proof is correct, because it's just making a lot of assumptions and stuff. Proofs with prime numbers are practically impossible, except to prove that there are an infinite number of prime numbers. And a lot of proofs are for crazy stuff that nobody cares about.

Edit:

http://mathworld.wolfram.com/EuclidNumber.html

This is the proof that the article is originally talking about. If anyone cares enough they can look up Euclid's Second Theorem, whatever that is. I don't really need a proof to believe that the number of primes is infinite though. If someone tells me then taht's good enough for me. Edited December 5, 2007 by cangor (see edit history)