Problem 6

Show that there exists a set

A

of positive integers with the following property: for any infinite set

S

of primes, there exist two positive integers

m\in A

and

n\notin A

, each of which is a product of

k

distinct elements of

S

for some

k\ge2

Step 4 of 4: Construct the non-member

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In plain words

Shifting the block changes its smallest prime factor but not its length.

n=s_2s_3\cdots s_{k+1}\notin A

Detailed analysis

Take $n=s_2s_3\cdots s_{k+1}$ . It is a product of $k\ge2$ distinct elements of $S$ , but its smallest prime factor is $s_2>s_1=k$ , while it still has $k$ factors. Hence $n\notin A$ , proving the required property.