Please prove the following:

1. There is an infinite number of highly composite numbers.

2. For any highly composite number (

**n= p**the

_{1}^{c1}*p_{2}^{c2}* p_{3}^{c3}*...p_{k}^{ck})**k**given prime numbers

**p**must be precisely the first

_{i}**k**prime numbers (

**i.e. 2, 3, 5,7,...**).

3. The sequence of exponents

**c**must be non-increasing.

_{k}4. Only in two special cases (which?) the last exponent

**c**is greater than 1.

_{k}Rem: Although number 1 does not exactly comply with my definition it is considered an HC number.