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Proof:
The proof is by contradiction.
Suppose there are only finitely many primes.
Let the complete list be $p_1,p_2,\dots,p_n$.
Let $N = p_1p_2 \dots p_n+1$.
According to the Fundamental Theorem of Arithmetic, $N$ must be divisible by some prime.
This must be one of the primes in our list. Say $p_k \mid N$.
But $p_k\mid p_1\dots p_n$, so $p_k\mid(N-p_1 \dots p_n) = 1$
Hence contradiction.

I don't see how this proof works. I understand that $N$ isn't necessarily prime, but I don't understand how it apparently must show that some primes weren't in our list. A number could be made of different powers of the given primes right?

Someone please explain.

Martin Sleziak
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    A small point. Euclid's actual proof is not by contradiction. It says: Given any (finite) list of primes, to construct a prime not in that list. – GEdgar Jun 21 '14 at 01:34
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    For some people, proof by contradiction is a capital sin, hence they rush to defend the honor of a long dead mathematician. What I don't like about the proof as it was presented to you is that it has the proof of the fundamental theorem of arithmetic as a prerequisite. If I was a teacher, I'd present the fundamental theorem of arithmetic after this one. – Robert Soupe Jun 21 '14 at 01:53
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    @Robert The proof does not use FTA, only the Lemma that every natural $> 1$ has a prime factor, which has a very simple proof by induction. A direct proof is preferred not only because it is constructive, but also because it is far less confusing to beginners, who frequently become hopelessly confused reasoning in hypothetical structures such as "$\Bbb Z$ with only finitely many primes". This proof by contradiction is one of the major sources of confusion in general level math forums (there were thousands of posts about such on sci.math). – Bill Dubuque Jun 21 '14 at 02:38
  • @RobertSoupe : I do not consider proof by contradiction wrong, but I consider it wrong to make this proof into a proof by contradiction, for reasons set forth in my answer below and in my published work on this proof. – Michael Hardy Jun 21 '14 at 02:48
  • BTW I agree that uniqueness of prime factorizations is not needed in this argument. Existence, on the other hand, is needed, of course. – Michael Hardy Jun 21 '14 at 02:53
  • @BillDubuque The way Exam in 5 Days phrased the question makes it sound like he was first taught the FTA and later he was taught the theorem on the infinitude of primes. If that's the case and I was in his shoes, I would feel like I have to completely understand the FTA before I can make sense of the infinitude of primes. – Robert Soupe Jun 21 '14 at 03:28
  • @MichaelHardy So you're the one who wrote that paper. The way you present the proof in your answer below is very neat and clean. But going back to the ancient Greek text seems kind of like overkill to me, like those musicologists who measure in millimeters Mozart's staccato strokes to try to distinguish dots from pikes. – Robert Soupe Jun 21 '14 at 03:34
  • @Michael No, existence is not used, only the Lemma that every integer $> 1 $ has a prime factor. Existence follows from that by induction, but it is not used above. One can even eliminate that lemma if one instead proves that there are infinitely many coprimes (the essence of the matter). – Bill Dubuque Jun 21 '14 at 03:36

8 Answers8

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Suppose that there are only $n$ primes. Let $p_1,p_2,p_3,\cdots,p_n$ be all the primes in the world. Let $N$ be the product of all these primes plus $1$, i.e. $N=p_1p_2 \cdots p_n+1$. We know that $N$ must be a product of primes. But the only primes in the world are $p_1,p_2,\cdots,p_n$. So one of these must be a factor of $N$. Since it doesn't matter which, let's say $p_1$ is a factor of $N$. Then we use $$ N=p_1p_2\cdots p_n +1 $$ to find that $$ N-p_1p_2\cdots p_n =1 $$ But $p_1$ divides the left side of the equality since $p_1$ is a factor of $N$ (from above) and it divides $p_1p_2\cdots p_n$ because it's part of the product.

But since it divides the left side it must divide the right side of the equality. But that means $p_1$ divides $1$. But that can't happen because $p_1$ has to be at least as big as $2$ - the smallest prime there is - and no number except $1$ and $-1$ divide $1$. Therefore, we have a contradiction. This means our assumption - that there are only $n$ primes, is wrong.

What we have done is used our primes $p_1,p_2,\cdots,p_n$ to make a number which needs a new prime not among the list $p_1,p_2,\cdots,p_n$. So for any $n$ primes you have, you can always make a number forcing you to need at least one more prime which lets you make another such number and so forth. So of course there are infinitely many primes.

mathematics2x2life
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  • To understand the other two answers, I needed to have read your answer, so I suppose that makes your answer the best answer. Thank you for the long post, I understand now. – Exam in 5 days Jun 21 '14 at 01:30
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    No, the primes are not infinite; rather the set of all primes is infinite. There is not one infinite prime among them, much less is it the case that all of them are infinite. At any rate this is an inferior proof. The way Euclid proved it, which was not by contradiction, is superior. – Michael Hardy Jun 21 '14 at 02:49
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    No problem. @MichaelHardy Thank you for correcting my loose language. When I said the primes are infinite, of course I was being colloquial and meant the set of all prime numbers. And I agree. Euclid's original constructive proof is simple and more informative than the proof by contradiction which changes little of his original proof. I can think of no other reason of why it's commonly taught other than someone thought of it once as an early-on example of proof by contradiction and people just stuck with it. Why they don't just stick with the irrationality of $\sqrt{2}$ is beyond me. – mathematics2x2life Jun 21 '14 at 19:09
  • @mathematics2x2life Can I just clarify what you meant by "Why they don't just stick with the irrationality of $\sqrt{2}$ is beyond me." Were you saying that they should use this as the primary example of proof by contradiction, or that $\sqrt{2}$ is somehow related to proving infinite primes.

    Second question: Why is proof by contradiction so ill-spoken? What makes it such a bad form of proof?

     – Exam in 5 days Jun 22 '14 at 07:45
  • @Examin5days The irrationality of $\sqrt{2}$ is often the introductory example to proof by contradiction and is a good, simple example. While proving the set of primes is infinite by contradiction is fine, it is unnecessary. Notice we can remove the assumption that the primes are finite and just assume there are $n$ of them. With the discussion above, we show using these $n$ primes we can make a number which needs a new prime not among those $n$, so there are at least $n+1$ primes. So the set of primes is infinite - no need for the contradiction. – mathematics2x2life Jun 22 '14 at 21:43
  • @Examin5days In general, when possible, there are groups of mathematicians that believe a constructive proof is always better and doing the above is a bit more constructive than the contradiction proof. Moreover, you see how the contradiction really isn't necessary. It doesn't change the proof and just eats up word space. That is what makes it 'bad'. But in general, proof by contradiction it a useful and nice form of proof. It's just here that it really isn't necessary. So I was just saying that the $\sqrt{2}$ proof is a better example for contradiction than this. – mathematics2x2life Jun 22 '14 at 21:45
  • Thank you for the clarification! – Exam in 5 days Jun 22 '14 at 23:30
  • The bit I have always got hung up on on this type of proof is the line, "We know that $N$ must be a product of primes", which is asserted in the proof, but it's not obvious to the untrained eye. A proof can be seen here: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic#Existence Then, the bit I get hung up on on the proof in the above link is the line, "Otherwise, there are integers $a$ and $b$, where $n = ab$, and $1 < a ≤ b < n$". Again, this is asserted but not justified... – Adam Rubinson Dec 03 '20 at 14:11
  • ...However, I now see that this comes from the definition of a prime number: a prime number is a number divisible only by $1$ and itself. If a number $n$ is not prime then $n$ is not only divisible by $1$ and itself: a little bit of deduction shows that there must exist two - not necessarily distinct - numbers $a,b$ such that $n =ab$ and neither $a$ nor $b$ are $1$ or $n$. – Adam Rubinson Dec 03 '20 at 14:11
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  • We have a list of primes
  • $N$ is divisible by a prime.
  • None of the primes in the list divides $N$

Therefore...

  • $N$ is divisible by a prime not on the list

and thus

  • There is a prime not on the list
9

This proof by contradiction is a badly organized proof. It should instead be done like this:

Suppose you have any finite set of primes $p_1,\ldots,p_n$. (DO NOT assume that there are no other primes.)

Then no prime factor of $(p_1\cdots p_n)+1$ can be one of the primes $p_1,\ldots,p_n$. (That part you can prove by contradiction without, as far as I know, introducing flaw into the proof that would not otherwise be there.)

Therefore there is at least one more prime than $p_1,\ldots,p_n$. No matter how many you've listed so far, there is at least one more. ${}\qquad\blacksquare$

That is how Euclid did it.

One of the problems of rearranging this into the frequently seen proof by contradiction is just that that adds an extra complication to the proof that serves no purpose, thereby making the proof appear more complicated than it really is.

Another problem is this: Some authors (e.g. G. H. Hardy (not related to me as far as I know)) say something along these lines: Because $(p_1\cdots p_n)+1$ is not divisble by any of the primes, it must itself be prime, and then we get a contradiction. Only the assumption that the list $p_1,\ldots,p_n$ contains all primes --- an assumption that is present only when this is rearranged into a proof by contradiction --- causes anyone to say that it is not divisible by any primes, and only that conclusion makes anyone say that therefore it is itself prime. This leads students to think mistakenly that it has been proved that if you multiply the first $n$ primes and then add $1$, the number you get is always prime. But that is false. And if a student then finds counterexamples (e.g. the six smallest primes) then the student may mistakenly conclude that the proof is simply wrong.

Another problem with preseting it as a proof by contradiction is that authors who do that often explicitly state that Euclid did it that way. That is historically false.

Catherine Woodgold and I wrote a joint paper, published in the Mathematical Intelligencer in fall 2009, debunking the false historical claims and demonstrating the superiority of Euclid's version over the proof by contradiction falsely attributed to Euclid.

And you shouldn't say "infinite primes" when you mean "infinitely many primes". "Infinite primes" would be primes each one of which is infinite. In colloquial speech the word "infinite" may be used that way, but in mathematical terminology it is an incorrect usage.

Michael Hardy
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    All of that (and more!) was done long before said paper was published in many diverse threads on sci.math (some of which M.H. participated in). – Bill Dubuque Jun 21 '14 at 03:47
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    But how is the proof given not a proof by contradiction? I.e. to deduce "Set of all primes is infinite" from "No matter how many you've listed so far, there is at least one more" one needs some contradiction argument or not? [At least if we define a set to be infinite iff there is no bijection to some set of the form ${1,...,n}$] – Andrei Kh Sep 13 '17 at 12:27
  • @AndreiKh : It doesn't start with the assumption that only finitely many primes exist. – Michael Hardy Sep 13 '17 at 17:31
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    @MichaelHardy Yes, but to get the final statement "Set of all primes is infinite" one needs some contradiction. Or am I missing something obvious? – Andrei Kh Sep 13 '17 at 20:25
  • @AndreiKh : Perhaps it depends on your definitions. One of Euclid's translators stated the theorem thus: "Prime numbers are more than any assigned multitude of prime numbers." – Michael Hardy Sep 13 '17 at 20:30
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Hint $\ $ Consecutive integers are always coprime: $\, d\mid N,N\!-\!1\,\Rightarrow\, d\mid N-(N\!-\!1) = 1.$

Thus $\,N\,$ is coprime to $\,N\!-\!1 = p_1\cdots p_n\,$ so $\,N\,$ is not divisible by any of the primes $\,p_i.$

But $\,N>1\,$ so it has some prime factor $\,p.\,$ By above $\,p\,$ is a "new" prime, not among the $\,p_i.$

Remark $\ $ Euclid's proof was constructive like the above, i.e. not by contradiction.

Bill Dubuque
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$p_1$ can't divide $N=p_1(\cdots)+1$ since if you divide $N$ by $p_1$ you get a remainder of 1. The same holds for each of the primes so none can divide $N$, hence there is some other prime.

Charles
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Here is another idea to prove this statement. Our instructor attributed to the french mathematician Hermite. The proof goes as follows:

for a given integer $n$. Let $q_n$ be the greatest prime factor of $n! +1$. We claim that $q_n > n$. If not so, then clearly $q_n | n! $ and by construction $q_n|(n! + 1)$, consequently $q_n|1$. Which is impossible, a contradiction reached. It follows that always there exists a prime number $q_n$ greater than $n$ for any integer $n$ as required.

Maged Saeed
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  • While I like this approach, the original question is more about clarification in a specific proof. Often it is okay to propose alternative proof in such situations, but in this case I would say there are more suitable questions where to post this as an answer, such as Different ways to prove there are infinitely many primes? (provided it is not already there). Also as a side effect, that question has wider audience than this one, which is always better :) – Sil Sep 20 '18 at 07:09
  • @Sil very thankful for your delighting comment. I am going to put it there. – Maged Saeed Sep 20 '18 at 07:55
  • @Sol.I cannot add any more answer to the question you refered to. Why? – Maged Saeed Sep 20 '18 at 08:04
  • Oh, I realized why. I have only one reputation in this site omitting the bonus points from other sites in the network. If I got 9 more reputation, equally to an upvote, I, then, can add an answer there. :) – Maged Saeed Sep 20 '18 at 08:28
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Can this be considered a proof? why or why not

Let's say the last prime number we know is Pn, then there has to exist a prime number Pn+1 because Pn * Pn+1 results to a number N whose prime factorisation is Pn and Pn+1. Since numbers are infinite, N exists and so does Pn and Pn+1

ffff
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In this case, I assume there will be a superior prime, which if every prime existed is odd, then there exist $k+1$ if there exist $2k+1$, for some $k$ of course.

The proof seems not by contradiction, but by superiority, assuming that all products by prime is prime, in this case, $N+1$ that is divisible by prime number, which is false. Since, there will always a reminder of 1, the proof seems to be flawed.

However, if you consider that, every prime number is odd, in this case, $（2k+1）$ for $N$, and I will prove in this way,

$N = （2k+1）* （2k+1） .... + 1$ for some prime numbers, which lead to the factor of 2, and thus, leading it into the divisibility of 2, thus, reducing it to k+1 for some primes, that for me already proved, all primes are odd when $>2$, which $2k+1$, and have the smallest prime of 2, when $k+1$. Since, k, is some numbers that satisfy prime, k cannot be an empty set. It proves that k+1 and 2k+1, is uniquely existed, thus, k cannot be zero. Thank you

Especially Lime
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Alyaa Ashraf
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