A highly composite number (HCN) is a positive integer having more divisors than any smaller positive integer (sequence A002182 in OEIS).
Please prove the following:
1. There is an infinite number of highly composite numbers.
2. For any highly composite number (n= p1c1*p2c2*
the k given prime numbers pi must be precisely the first k prime numbers ( i.e. 2, 3, 5,7,...).
3. The sequence of exponents ck must be non-increasing.
4. Only in two special cases (which?) the last exponent
ck is greater than 1.
Rem: Although number 1 does not exactly comply with my definition it is considered an HC number.
1. First, see this wikipedia
for a number n=p^c with p prime and c an integer greater than 0. Then number of divisors is c+1. Thus, for any integer x>1 we can find an integer n>=1 with x divisors. And, because n is bounded below, there must be a minimum n. Thus, for this x, n would be an HCN. Thus the HCN's are infinite.
Posted by Daniel
on 2012-10-28 06:58:03