Understanding Fundamental Theorem of Arithmetic

In this class, We discuss Understanding Fundamental Theorem of Arithmetic.

The reader should have prior knowledge of prime factorization. Click Here.

Every integer N > 1 can be expressed uniquely as a product of prime numbers.

The proof is divided into two parts.

1) Existence

2) Uniqueness

Existence:

First step, N = 2

Two is a prime number

N = 3, 3 is a prime number.

N = 4, 4 can be expressed as 2 * 2.

N = 5, 5 is a prime number.

N = 6, 6 can be expressed as 2 * 3.

N = k+1

K+1 may be a prime number or composite number.

If K + 1 is the prime number, then there is no need to express it in primes.

If K + is a composite number, K + 1 can be expressed as UV.

U and V are numbers less than K + 1.

Example:

Number 32 can be expressed as 8 * 4.

The numbers 8 and 4 are already expressed in prime numbers.

We can take any number and express it as a product of prime numbers.

Uniqueness:

Any number can be expressed uniquely.

Thirty-two can be expressed in only one way: 2^5.

Assume K + 1 can be expressed in two ways.

K + 1 can be expressed as p1, p2, p3, . . pn and q1, q2, q3, . .qm

P1 divides K + 1

P1 divides q1, q2, q3, . .qm

P1 = qk because both are prime numbers.

So we can express ourselves only in one way.