I have a solution but I am confused why this book uses more abstruse language or I am missing something (as in are my solutions not rigorous enough):

Find the least positive integer n such that any set of $n$ pairwise relatively prime integers greater than $1$ and less than $2005$ contains at least one prime number. The solution from Putnam and Beyond uses contradiction and least/maximums of prime factorizations. I think the solutions are basically the same, but the Putnam one has me a bit confused.

Here's mine:

Since the set of positive integers is relatively prime, each positive integer is comprised of the primes $2,\dots,43$. Considering a set of no primes, we can use powers of these primes to construct the set, the largest set of which being the squares of the primes from $2,...,43$ which has $14$ elements.

In order for the set to remain pairwise relatively prime any new integer appended to the set must have be divisible by a prime greater than $43$. The smallest being $47$. Since the set must be comprised of composite numbers, the smallest number that satisfies this property is $47^2$ which is greater than $2005$. So any such set with at least $n = 15$ elements must have at least one prime number and from above it is seen that we can construct sets that have pairwise relatively prime integers and are composite for $n < 15$ by considering n squares of primes. Hence $n = 15$.

  • $\begingroup$ Your first sentence this is only true if the set doesn't contain primes (like, say, $101.$) So you are implicitly using proof by contradiction. Why can't $202$ be in your set? Also, break your answer into paragraphs. Very hard to read one long paragraph. $\endgroup$ – Thomas Andrews Jun 27 '19 at 19:28
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    $\begingroup$ You've shown that one set of $14$ relatively prime and non-prime values cannot be extended to $15.$ But .that doesn't show it for all sets of $14$ non-prime relatively prime values. $\endgroup$ – Thomas Andrews Jun 27 '19 at 19:34
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    $\begingroup$ The basic argument is that if $m$ is not prime and in $2,\dots,2004,$ then $m$ is divisible by a prime in $2,\dots,43.$ $\endgroup$ – Thomas Andrews Jun 27 '19 at 19:37
  • $\begingroup$ Your intuition is correct, but there are other sets. For example: $\{2p_{27},3p_{26},5p_{25},\cdots,41p_{15},43^2\}$ is another example of 14 non-primes relatively prime. $\endgroup$ – Thomas Andrews Jun 27 '19 at 19:53

You've shown that one set of $14$ relatively prime and non-prime values cannot be extended to $15.$ But that doesn't show it for all sets of $14$ non-prime relatively prime values. Another $n=14$ example set is $\{2p_{27},3p_{26},5p_{25},\cdots,41p_{15},43^2\},$ where $p_{15}=47,p_{16}=53,\dots,p_{27}=103.$

Your instinct is correct, but it is just not a complete proof.

This is a neat problem, because the intuition feels instinctive, but proving it correctly requires some technique.

Given any $m>1$, define $d(m)$ to be the smallest prime divisor of $m.$

Claim 1: If $2\leq m\leq 2004$ is not prime, $d(m)\leq 43.$

Proof: Otherwise, $m\geq p_1p_2$ for some pair of primes $p_1,p_2$ and $p_i\geq 47.$ But then $m\geq 47^2=2209.$

Claim 2: If $m_1,m_2>1$ are relatively prime, then $d(m_1)\neq d(m_2).$

Proof: If $p=d(m_1)=d(m_2)$ then $p$ us a common factor of $m_1$ and $m_2.$

Claim 3: Given any set $S=\{m_1,\cdots,m_{15}\}$ of non-prime values with $2\leq m_i\leq 2004.$ Then the set is not pairwise relatively prime

Proof: There are at most $14$ distinct possible values of $d(m_i)$ by claim $1,$ since there are $14$ distinct primes $\leq 43$. Thus $d(m_i)=d(m_j)$ for some $i\neq j.$ Then by claim $2,$ $m_i$ and $m_j$ are not relatively prime.

Your example of the squares $S=\{2^2,\cdots,43^2\}$ shows that $n=15$ is the smallest such $n$ for which it is true.

  • $\begingroup$ Your fourth comment helped a lot, thanks. $\endgroup$ – Derek Luna Jun 27 '19 at 20:32
  • $\begingroup$ I was confused in thinking I was only going to be able to use primes $\le$ 43... $\endgroup$ – Derek Luna Jun 27 '19 at 20:42

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