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In my calculus textbook we are asked to prove that every integer greater than 1 is a product of primes. This theorem is not new to me, however, the proof they provide seems unnecessarily long.

Proof: The induction step holds because 2 is prime and thus also a product of primes. Let's assume that every number in the set $2,3,4,...,n$ is a product of primes and we want to prove that $n+1$ is also a product of primes. Let $p$ be the smallest number from $2,3,4,...,n,n+1$ that divides $n+1$. We see that $p$ is prime, because if it's not there exists a $d$ such that $2 \le d<p$ that divides $p$ and also divides $n+1$ but $p$ is minimal a contradiction, therefore $p$ is prime. If $p=n+1$ then $n+1$ is prime and we are done. Else, $p<n+1$, and $q=(n+1)/p$ is bigger than 1 and smaller than $n+1$, and therefore from the induction hypotheses $q$ is the product of primes. Thus, $n+1=pq$ is also a product of primes.

I don't understand why $p$ needs to be prime. Shouldn't it be enough to know that $p$ is a product of primes and then we just get to the conclusion much faster?

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    As you suggest, there are many ways to rewrite this argument. A standard one is to say that either $n+1$ is prime or it isn't...if it is we are done, if it isn't then $n+1=a\times b$ where we already know that $a,b$ are products of primes (as they are both in ${2, \cdots, n}$), so we are done. – lulu Nov 28 '22 at 12:51
  • In other words, we use strong induction. – Peter Nov 28 '22 at 13:02

2 Answers2

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Staying with the method used, during the strong-induction hypothesis, every integer $k$ such that $~2 \leq k \leq n~$ is a product of primes. This does not immediately imply that $(n+1)$ is a product of primes.

If $p$ is the smallest number that divides $(n+1)$, and $p$ happens to be less than $(n+1)$, then you can conclude that both $p$ and $~\dfrac{n+1}{p}~$ are the product of primes.

However, you could have that $p = (n+1),$ rather than $p < (n+1).$ In this situation, the analysis in the previous paragraph does not hold, so you can not then immediately conclude that $(n+1)$ is the product of primes.

It is only after concluding that in this exceptional case, that $p = (n+1)$ must therefore be prime, and thus the product of primes, can you construe that the strong-induction hypothesis step has been completed.

user2661923
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  • Why can't we just say either $n+1$ is prime(in which case we're done) or $n+1$ is composite and therefore $n+1=ab$ where $a,b<n+1$ and therefore $n+1$ is a product of primes? –  Nov 28 '22 at 13:30
  • @DGJXDGF4564 In effect, that is the same analysis as is used in the given proof. The distinction is that in the given proof, elegance was dispensed with, in favor of one slow careful step at a time. – user2661923 Nov 28 '22 at 13:31
  • So what step in the shorter proof could be better explained? What assumptions am I making that the more careful proof didn't? –  Nov 28 '22 at 13:37
  • @DGJXDGF4564 None. The shorter proof is fine. It is simply that for some students, brand new to the topic, their intuition will be so unsophisticated that they will have to stop and think to realize that the shorter proof is (also) valid. It is not uncommon for proofs that are more long-winded to be more easily grasped by the brand-new Math students. – user2661923 Nov 28 '22 at 13:39
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Yes, if the least factor $\,p>1\,$ of $\,n\!+\!1$ is $\rm\color{#c00}{smaller}$ $(p < n\!+\!1)$ we do not need to prove $\,p\,$ is prime, since then ${\rm\color{#c00}{induction}}\Rightarrow p$ is a product of primes. But if $\,p = n\!+\!1\,$ then induction does not apply so then we need to use the knowledge that $\,p\,$ is prime to conclude the proof.

We can elmininate this inelegant exceptional case by instead basing the induction not at $\,n=2\,$ but instead at $\,n=1,\,$ viewing $1$ as an empty product of primes. Now the inductive step works smoothly: $ $ the least factor $\,p>1\,$ of $\,n\!+\!1\,$ is prime (as they proved) and $\,(n\!+\!1)/p < n\!+\!1,\,$ so induction $\Rightarrow (n\!+\!1)/p\,$ is a product of primes (possibly empty), so appending '$p$' to this product yields a prime factorization of $\,p(n\!+\!1)/p = n\!+\!1,\,$ completing the induction.

More generally and naturally: a simple induction shows that if a set of positive naturals contains $1$ and all primes and is closed under multiplication, then it is the set of all positive naturals. But the set of positive naturals writable as prime products clearly satsifies all those hypotheses, hence it is includes every positive natural.

This method of proof is more faithful to the (algebraic) structural heart of the matter - that the multiplicative monoid of positive naturals is (freely) generated by the primes, so that above method of induction is the natural structural induction to apply in multiplicative contexts (i.e. induction proofs "piggyback" on the inductive geenration of the structure). Follow the prior link for much further discussion.

Bill Dubuque
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  • So is there a reason why they made $p$ prime when $p<n+1$? –  Nov 28 '22 at 16:00
  • @DGJ That's difficult to infer w/o further context. It can be skipped as in my linked post, i.e. if $,n!+!1>1,$ is a prime $,p,$ then it has prime factorization '$p$', else $,n!+!1,$ is composite so it has a nontrivial factor $,1 < a < n!+!1,$ so by induction $,a,$ and $,(n!+!1)/a,$ have prime factorizations, which appended yield a prime factorization of $,n!+!1.\ \ $ – Bill Dubuque Nov 28 '22 at 17:31
  • @DGJ E.g. the given proof might be easier to follow based upon previously presented examples of induction in the textbook, esp. if algorithmic, i.e. a constructive interpretation of the proof yields a recursive algorithm for prime factorization: find the least prime factor $,p,$ of $,n,$ then apppend that to the prime factorization of $,n/p.,$ It may be easier for beginners to understand how that works (vs. the algorithm in my prior comment, which chooses any factor $,a,$ of $,n,$ & recurses on both factors $,a,$ & $,n/a\ \ $ – Bill Dubuque Nov 28 '22 at 17:47