Are there infinite numbers with four or six factors?
By Ali Korotana (P4195) on Monday, May 7, 2001 - 08:07 pm :

There exists a way of showing that there are infinite numbers with two factors, i.e. prime numbers. Is there a way of showing that there are infinite numbers with four or six or however many factors? I ask since we are looking for values of n where (n!)2 -1 has four factors.

By Dan Goodman (Dfmg2) on Monday, May 7, 2001 - 08:17 pm :

Sure, if p1 =2, p2 =3, ... are the prime numbers then pi pj (where i is not equal to j) has 4 factors: 1, pi , pj , pi pj . If you want 6 factors, try pi 2 pj (where i is not equal to j), since the factors are 1, pi , pi 2 , pi pj , pj , pi 2 pj . And so on...

By Ali Korotana (P4195) on Monday, May 7, 2001 - 08:28 pm :

Thats quite interesting but I'm not sure I'm really understanding why it 'works', could you elaborate a bit please Dan?

By Dan Goodman (Dfmg2) on Monday, May 7, 2001 - 10:01 pm :
Ali, do you know about unique factorisation? Every integer can be written uniquely as a product of prime numbers. Once you've got this, it's reasonably easy to see that the factors of, say, pi2 pj are what I said and that there are no others. In fact, there are an infinity of numbers with n factors if n > = 2. See if you can extend the method I used for 4 and 6 factors to work for n factors by writing n as a product of primes. It's quite tricky so if you can't work it out after a bit I'll post the solution. The key fact you need to use is that all of the factors of p1n1p2n2¼ pknk are just the numbers p1m1p2m2¼pkmk where each mi £ ni.
By Dan Goodman (Dfmg2) on Monday, May 7, 2001 - 10:42 pm :
D'oh! Just realised that there's a very easy proof of the problem I set above. Still, I'll leave it unsolved to see if you can get it.

To make it a bit more difficult for you, try and prove that if n=p1n1¼pknk then there is an infinity of numbers r with n factors such that the number of prime factors of r is n1+¼+nk. Hence prove that there is an infinity of numbers r with n factors such that the number of prime factors of r is ³ log2 n.