This post discusses and proves the fundamental theorem of arithmetic.
Finding the prime factors of a large integer is no easy feat. For example, the ninth Fermat number has 155 digits and has the following three prime factors:
These prime factors have 9 digits, 49 digits and 99 digits, respectively. They were published in 1993 after lengthy calculations involving approximately 700 workstations and a supercomputer using a number field sieve algorithm (see ). If the project were to be done today, it could certainly be done much faster and more efficiently with the current state of the art in supercomputing. However, the project will still be no trivial feat.
The following 617-digit integer is a product of two prime numbers and has yet to be factored by anyone (or any computer or any sets of computers). According RSA Laboratories, barring fundamental algorithmic or computing advances, the above number or other similarly sized number is expected to stay unfactored in the decades to come. For more background about this large number, see see RSA challenge number and RSA factoring challenge.
Though the implementation of prime factorization may be difficult or even impossible for large numbers, the fundamental theorem of arithmetic guarantees that any positive integer can be expressed uniquely as a product of prime numbers. The fundamental theorem of arithmetic is an essential theorem in number theory. In this post, we present of proof of this fundamental theorem.
The Fundamental Theorem of Arithmetic
The theorem has two parts – the existence and the uniqueness of the prime factorization. The existence part is relatively easy. We first prove the existence part. We use Euclid’s lemma to prove the uniqueness part.
Any integer has a prime divisor.
Let be an integer with . If is prime, then it has a prime divisor, namely . So assume is not prime. Then for some positive integers and smaller than .
Now consider the set of all positive integer divisors of . The set is nonempty since and belong to this set. The set is also a finite set since an integer can only have finitely many integer divisors. Every finite set of real numbers has a smallest element. Let be the least element of .
The must be a prime number. If not, would have a positive divisor that is smaller than . Then is also a positive divisor of , contradicting that is the least element of .
Any integer is a product of prime numbers.
We prove by induction. The lemma is certainly true for . Suppose the lemma is true for all positive integers where .
If is prime, then we are done. Assume is not prime. Then for some positive integers and smaller than . By induction hypothesis, both and are product of prime numbers. We can conclude that is also a product of prime numbers.
If is a prime number and , then or .
Proof of Euclid’s Lemma is found in this post. As a corollary of Euclid’s Lemma, we have the following lemma.
If is a prime number and , then for some .
We prove by induction. The case for is the Euclid’s Lemma. Assume that the lemma is true for where . Suppose that . By Euclid’s lemma, we have or . In the first case, the induction hypothesis tells us that for some with . In the second case . The two cases together tell us that for some with .
Since the validity of the lemma for implies the validity of the lemma for and since the lemma is true for , we now know that the lemma is true for all integers . This establishes the lemma.
Fundamental Theorem of Arithmetic
Any interger can be expressed uniquely as a product of prime numbers.
Let be an integer with . The existence of the prime factors of is established by Lemma 2. We now show there is only one prime factorization for . We would like to point out that order of the prime factors is not important. For example, we consider and as the same factorization for the integer .
Suppose that and are prime factorizations of the integer . We show that each is identical to some and each is identical to some . This implies that the prime numbers are simply a rearrangement of such that the only difference for the two factorizations is in the order of the factors.
Note that . Thus divides , or we write . By Lemma 3, for some . Since they are prime, . Then cancel out on both side of the equation, we have
Note that divides the right-hand side of the above equation. By Lemma 3, for some . Continue in this same manner, we see that each is identical to some . Furthermore, . Otherwise, there would be some left after we cancel out all the , leading to the situation that the product of several prime numbers is equal to one.
By interchanging with , the same argument will also show that each is identical to some . Furthermore, . Thus we have and that the two factorizations are really just rearrangement of each other.
From the fundamental theorem of arithmetic, it follows that any positive integer greater than one can be expressed uniquely in the following way:
where is a positive integer and each . Such representation of an integer is called a prime-power decomposition. For example, .
- Lenstra, A. K., Lenstra, H. W., Manasse, M. S., Pollard, J. M. The Factorization of the Ninth Fermat Number, Mathematiics of Computation, Volume 61, Number 203, July 1993, Pages 319-349.