# Why is the fundamental theorem of arithmetic special?

• japplepie
In summary, the significance of primes being fundamental lies in their role as the building blocks of positive integers. While it is possible to build up all natural numbers using addition, the use of primes in multiplication is more efficient. This is due to the fact that multiplication and addition are the fundamental operations used to define other operations, making them essential in understanding the structure of numbers. The name "fundamental theorem of arithmetic" may have been bestowed upon it by tradition, but ultimately it serves as a helpful label for this concept.

#### japplepie

Why is it significant enough to be fundamental?

Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.

Yes, you can build up all natural numbers by adding 1's. The primes do the same thing from the point of view of multiplication as 1 does from the point of view of addition. If you want to build up all the natural numbers using multiplication, you need to use primes.

In that case, why is multiplication so special? What makes it more special than addition, exponentiation, tetration, pentation, hexation, ... ?

japplepie said:
In that case, why is multiplication so special? What makes it more special than addition, exponentiation, tetration, pentation, hexation, ... ?
Multiplication is certainly not more "fundamental" than addition, but building up the natural numbers by adding 1's is apparently not difficult enough to call it a theorem. It may be how the natural numbers are defined. Multiplication and addition are the fundamental operations that are used to define the other operations.

If the question has to due with the fact why it has the name it has. I imagine this is a tradition bestowed upon it by the fact that the theorem has existed since Euclid. I'm sure there may exist other reasons. Anyway, I wouldn't get to hung up on the name. Sometimes we just name things so everyone knows what we're talking about.

## 1. Why is the fundamental theorem of arithmetic considered special?

The fundamental theorem of arithmetic is considered special because it is a fundamental concept in number theory that states every positive integer can be expressed as a unique product of prime numbers. This means that every positive integer has a unique prime factorization, making it a key building block for understanding the properties of numbers.

## 2. How does the fundamental theorem of arithmetic relate to prime numbers?

The fundamental theorem of arithmetic relates to prime numbers because it states that every positive integer can be broken down into a unique product of prime numbers. This means that prime numbers are the basic building blocks of all positive integers, and without them, the fundamental theorem of arithmetic would not hold true.

## 3. Can the fundamental theorem of arithmetic be proven?

Yes, the fundamental theorem of arithmetic can be proven. It was first proven by the Greek mathematician Euclid and has since been proven using different methods by other mathematicians. The most common proof involves using the principle of mathematical induction.

## 4. What practical applications does the fundamental theorem of arithmetic have?

The fundamental theorem of arithmetic has numerous practical applications. It is used in cryptography to ensure secure communication, in coding theory to detect and correct errors in data transmission, and in computer science for efficient algorithms. It also has applications in fields such as chemistry, physics, and biology.

## 5. Is the fundamental theorem of arithmetic unique to positive integers?

Yes, the fundamental theorem of arithmetic is unique to positive integers. It does not hold true for negative numbers, fractions, or decimals. However, there are similar theorems for other number systems, such as the unique factorization theorem for polynomials.