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 3 of 4: Construct the member of

A

Previous step Next step

In plain words

The first block has exactly as many factors as its smallest prime factor.

m=s_1s_2\cdots s_k\in A

Detailed analysis

Take $m=s_1s_2\cdots s_k$ . Its prime factors are distinct, its smallest prime factor is $s_1=k$ , and it has exactly $k$ factors. Therefore $m\in A$ .