Page 54 - tmp

Theory of highly-composite numbers

Dejan Kovač
Mentor mr Stevan Kordić
High school “Ruđer Bošković”, Belgrade/Serbia

1 Introduction Theorem 2 Let be highly-composite.

Then .

Ramanujan defined highly-composite number as a natural

number which is a “recorder” in number of divisors, Theorem 3 Let be highly-composite. If

meaning it has more divisors than any other smaller natural , then .
number. This research investigates the theory of highly-

composite numbers. Theorems of similar content haven’t Theorem 5 Let be highly-composite. If

been found in literature. After the research, a generalization and , then

of constraints and conditions that exponents in highly-

composite number have to satisfy has been made. Twenty- Theorem 6 Let be highly-composite. If

two theorems have been formulated together with their , then .

proofs, which give the limitations of exponents in prime

numbers in a highly-composite number. The formulated Theorem 7 Let be highly-composite. If

theory can be used as a theoretical background in , then

generating highly-composite number on computer systems.
Theorem 8 Let be highly-composite. If

2 Number theory , then .

Theorem 10 Let be highly-composite. If
Every positive number can be written as , where

and represent factors or divisors of number . Positive , then

number is defined as prime if it only contains 1 and Theorem 12 Let be highly-composite. If

as factors. If a number is not prime, than it is said to be

composite. The following things are also known: , then
a) Every composite number has it’s prime factors; Theorem 13 Let be highly-composite. If

b) There is not a largest prime number. , then

Every natural number can be represented in a unique way in
canonical form as:

Theorem 14 Let be highly-composite. If

where and are ordered prime factors of number then

for . In the rest of paper, it is assumed that Theorem 16 Let be highly-composite. If

, then

For a number written in canonical form, the number of
divisors of observed number is calculated in a following
way: Theorem 17 Let be highly-composite. If

. , then

It is important to note that there are infinitely many
numbers with same number of divisors, but since the set of Theorem 18 and 19 Let be highly-

natural numbers is well-ordered, it is possible to find the composite. If , then

smallest number containing given number of divisors. Besides that, also

This research also used Bertrand’s postulate which states
that: Theorem 21 Let be highly-composite.

a) If , then

Where and are two consecutive primes. b) If , then

c) If , then

3 Theory of highly-composite numbers d) If , then

e) If , then

Some of the theorems showed in research are:
Theorem 22 Let be highly-composite.

Theorem 1 Let be the smallest number Denote with the index of the prime number which

with divisors. Then: satisfies . Then .
a) are ordered prime numbers starting from 2,

meaning

b) .