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 1 of 4: Define the set

Previous step Next step

In plain words

Membership is controlled by an equality between the smallest prime factor and the number of factors.

A=\{q_1q_2\cdots q_{q_1}:q_1<q_2<\cdots<q_{q_1}\text{ are primes}\}

Detailed analysis

Let $A=\{q_1q_2\cdots q_{q_1}:q_1<q_2<\cdots<q_{q_1}\text{ are primes}\}$ . Thus a squarefree integer is in $A$ exactly when its number of prime factors equals its smallest prime factor. Every such number uses at least $q_1\ge2$ factors.