Another look at the fundamental theorem of arithmetic

This is the first post of a series of blog posts on the Chinese remainder theorem, often abbreviated CRT. In this post, we take another look at the fundamental theorem of arithmetic. This post is background for the next post, which will show that CRT can be built step by step from the fundamental theorem of arithmetic.

Links to the other posts in the series:
second post, third post, fourth post.

Every integer can be factored into a product of primes in essentially one way. This fact is called the fundamental theorem of arithmetic. For example, the number 84 is 2 x 2 x 3 x 7. The ordering of the primes is not important here since, for example, 2 x 2 x 3 x 7 is the same as 3 x 2 x 7 x 2. The example of 84 seems to suggest that the fundamental theorem of arithmetic is simply an exercise at the elementary school. In general, factoring a number is a very hard problem. For integers with hundreds or thousands of digits, finding the factors may take more seconds than the number of atoms in the universe! The fundamental theorem of arithmetic guarantees that such factorization exists for any integer, large or small. Because of this, the prime numbers are the building blocks of the integers (the atoms of arithmetic).

What is the importance of knowing that every integer can be factored into prime numbers in a unique way? A short answer is that the fundamental theorem of arithmetic is a foundation result. A vast array of mathematical structures are built on this foundation. The fundamental theorem of arithmetic is the beginning point of many mathematical stories. In this and subsequent posts, we highlight one such example. We show that the Chinese remainder theorem follows from the fundamental theorem of arithmetic. CRT combines beauty and utility. Even though CRT has been known for at least 1,000 years, it continues to present new applications in many areas, e.g. in coding theory, cryptography and the theory of computing. Surprisingly, the proof of CRT is quite simple. CRT follows from a lemma that is equivalent to the fundamental theorem of arithmetic. As a result, the discussion in this and the subsequent posts is a clear demonstration of the power of the fundamental theorem of arithmetic.

In this post, we discuss the fundamental theorem of arithmetic. In the next posts, we discuss CRT.

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The lemma

The starting point is the following lemma.

Lemma A
Let a, b and d>0 be integers. Suppose that \text{GCD}(a,d)=1, i.e. the greatest common divisor of a and d is 1. If d \lvert (a \cdot b), then d \lvert b.

Proof
Suppose that \text{GCD}(a,d)=1. By the extended Euclidean algorithm, the linear diophantine equation ax+dy=1 is solvable in integers. Let x and y be integers that satisfy this equation. Multiply this equation by b to obtain abx+dby=b. Since d \lvert (a \cdot b), d divides both terms on the left hand side of the last equation. Thus d \lvert b. \square

Lemma A is a simple statement. The proof is also simple, simply using the extended Euclidean algorithm, which is the Euclidean algorithm working backward. the Euclidean alogorithm consists, at heart, of a series of divisions to derive the greatest common divisor. As discussed below, the simple Lemma A leads to the fundamental theorem of arithmetic and Lemma A can also be derived from the fundamental theorem of arithmetic.

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The fundamental theorem of arithmetic

The fundamental theorem of arithmetic states that any positive integer greater than 1 can be expressed as a product of primes and that the factorization is unique. The theorem is proved in this previous post. Here, we show that it is equivalent to Lemma A.

Theorem B
The following conditions are equivalent:

  1. The statement of Lemma A.
  2. (Euclid’s lemma) If p is a prime number and p \lvert (a \cdot b), then either p \lvert a or p \lvert b.
  3. (Fundamental Theorem of Arithmetic) Every positive integer n>1 can be written as a product of prime numbers and that this product (called a factorization) is unique.

Condition 2 is usually referred to as the Euclid’s lemma. Condition 3 is, of course, a statement of the fundamental theorem of arithmetic. The story we would like to tell is that Lemma A is equivalent to the fundamental theorem of arithmetic. Thus any consequence of Lemma A is also a consequence of the fundamental theorem of arithmetic.

The proof of 1 \rightarrow 2 \rightarrow 3 is done in this previous post. We also like to point out one additional argument is needed to establish the equivalence of the three conditions. The proof to establish the fundamental theorem of arithmetic in the previous post uses an induction argument to show that every integer is a product of prime numbers. So condition 2 plus the induction argument establish condition 3. The role of condition 2 is to establish the uniqueness of the product of primes. We only need to show 3 \rightarrow 1.

Proof
3 \rightarrow 1
Suppose that a and d have no prime factors in common and that d \lvert (a \cdot b). So we have a \cdot b=d \cdot m for some integer m. By condition 3, we can express a and d as a product of primes as follows:

    \displaystyle p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_t^{\alpha_t} \times b=q_1^{\delta_1} \cdot q_2^{\delta_2} \cdots q_r^{\delta_r} \times m

The numbers p_i are the prime factors of a and the numbers q_i are the prime factors of d. The exponents \alpha_i and \delta_j are positive integers. Note that p_i \ne q_j for any i and j. Each q_i^{\delta_i} must appear in the prime factorization of the left-hand side. Since q_i^{\delta_i} cannot appear in the factorization of a, it must be in the factorization of b. \square

The loop 1 \rightarrow 2 \rightarrow 3 \rightarrow 1 shows the equivalence of the three statements. From a logical standpoint, any one of these statements is the fundamental theorem of arithmetic. As a demonstration of the power of the fundamental theorem of arithmetic, the next post shows that the Chinese remainder theorem is derived using Lemma A.

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\copyright \ 2015 \text{ by Dan Ma}

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3 thoughts on “Another look at the fundamental theorem of arithmetic

  1. Pingback: Versions of the Chinese remainder theorem | Exploring Number Theory

  2. Pingback: How to Chinese remainder, part 1 | Exploring Number Theory

  3. Pingback: How to Chinese remainder, part 2 | Exploring Number Theory

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