Fundamental theorem of arithmetic

History

Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid's Elements are essentially the statement and proof of the fundamental theorem.

If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.

Proposition 30 is referred to as Euclid's lemma. And it is the key in the proof of the fundamental theorem of arithmetic.

Any composite number is measured by some prime number.

Proposition 31 is proved directly by infinite descent.

Any number either is prime or is measured by some prime number.

Proposition 32 is derived from proposition 31, and prove that the decomposition is possible.

If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it.

Book IX, proposition 14 is derived from Book VII, proposition 30, and prove partially that the decomposition is unique – a point critically noted by André Weil. Indeed, in this proposition the exponents are all equal to one, so nothing is said for the general case.

Article 16 of Gauss' Disquisitiones Arithmeticae is an early modern statement and proof employing modular arithmetic.

Canonical representation of a positive integer

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

where p₁ < p₂ < ... < p_k are primes and the α_i are positive integers. This representation is commonly extended to all positive integers, including one, by the convention that the empty product is equal to 1 (the empty product corresponds to k = 0).

This representation is called the canonical representation of n, or the standard form of n.

Note that factors p⁰ = 1 may be inserted without changing the value of n (e.g. 1000 = 2³×3⁰×5³).
In fact, any positive integer can be uniquely represented as an infinite product taken over all the positive prime numbers,

where a finite number of the n_i are positive integers, and the rest are zero. Allowing negative exponents provides a canonical form for positive rational numbers.

Arithmetic operations

The canonical representation, when it is known, allows for a simple representation of the products, gcd, and lcm of two numbers:

However, as Integer factorization of large integers is much harder than computing their product, gcd or lcm, these formulas have, in practice, a limited usage.

Arithmetical functions

Many arithmetical functions are defined using the canonical representation. In particular, the values of additive and multiplicative functions are determined by their values on the powers of prime numbers.

Proof

The proof uses Euclid's lemma (Elements VII, 30): if a prime p divides the product of two natural numbers a and b, then p divides a or p divides b.

Existence

We need to show that every integer greater than 1 is either prime or a product of primes. For the base case, note that 2 is prime. By induction: assume true for all numbers between 1 and n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = ab and 1 < a ≤ b < n. By the induction hypothesis, a = p₁p₂...p_j and b = q₁q₂...q_k are products of primes. But then n = ab = p₁p₂...p_jq₁q₂...q_k is a product of primes.

Uniqueness

Assume that s > 1 is the product of prime numbers in two different ways:

We must show m = n and that the q_j are a rearrangement of the p_i.

By Euclid's lemma, p₁ must divide one of the q_j; relabeling the q_j if necessary, say that p₁ divides q₁. But q₁ is prime, so its only divisors are itself and 1. Therefore, p₁ = q₁, so that

Reasoning the same way, p₂ must equal one of the remaining q_j. Relabeling again if necessary, say p₂ = q₂. Then

This can be done for each of the m p_i's, showing that m ≤ n and every p_i is a q_j. Applying the same argument with the p 's and q 's reversed shows n ≤ m (hence m = n) and every q_j is a p_i.

Elementary proof of uniqueness

The fundamental theorem of arithmetic can also be proved without using Euclid's lemma, as follows:

Assume that s > 1 is the smallest positive integer which is the product of prime numbers in two different ways. If s were prime then it would factor uniquely as itself, so there must be at least two primes in each factorization of s:

If any p_i = q_j then, by cancellation, s/p_i = s/q_j would be another positive integer, different from s, which is greater than 1 and also has two distinct factorizations. But s/p_i is smaller than s, meaning s would not actually be the smallest such integer. Therefore every p_i must be distinct from every q_j.

Without loss of generality, take p₁ < q₁ (if this is not already the case, switch the p and q designations.) Consider

and note that 1 < q₂ ≤ t < s. Therefore t must have a unique prime factorization. By rearrangement we see,

Here u = ((p₂ ... p_m) - (q₂ ... q_n)) is positive, for if it were negative or zero then so would be its product with p₁, but that product equals t which is positive. So u is either 1 or factors into primes. In either case, t = p₁u yields a prime factorization of t, which we know to be unique, so p₁ appears in the prime factorization of t.

If (q₁ - p₁) equaled 1 then the prime factorization of t would be all q's, which would preclude p₁ from appearing. Thus (q₁ - p₁) is not 1, but is positive, so it factors into primes: (q₁ - p₁) = (r₁ ... r_h). This yields a prime factorization of

which we know is unique. Now, p₁ appears in the prime factorization of t, and it is not equal to any q, so it must be one of the r's. That means p₁ is a factor of (q₁ - p₁), so there exists a positive integer k such that p₁k = (q₁ - p₁), and therefore

But that means q₁ has a proper factorization, so it is not a prime number. This contradiction shows that s does not actually have two different prime factorizations. As a result, there is no smallest positive integer with multiple prime factorizations, hence all positive integers greater than 1 factor uniquely into primes.

Generalizations

The first generalization of the theorem is found in Gauss's second monograph (1832) on biquadratic reciprocity. This paper introduced what is now called the ring of Gaussian integers, the set of all complex numbers a + bi where a and b are integers. It is now denoted by Z [ i ] . He showed that this ring has the four units ±1 and ±i, that the non-zero, non-unit numbers fall into two classes, primes and composites, and that (except for order), the composites have unique factorization as a product of primes.

Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring Z [ ω ] , where ω = − 1 + − 3 2 ,   ω 3 = 1 is a cube root of unity. This is the ring of Eisenstein integers, and he proved it has the six units ± 1 , ± ω , ± ω 2 and that it has unique factorization.

However, it was also discovered that unique factorization does not always hold. An example is given by Z [ − 5 ] . In this ring one has

Examples like this caused the notion of "prime" to be modified. In Z [ − 5 ] it can be proven that if any of the factors above can be represented as a product, e.g. 2 = ab, then one of a or b must be a unit. This is the traditional definition of "prime". It can also be proven that none of these factors obeys Euclid's lemma; e.g. 2 divides neither (1 + √−5) nor (1 − √−5) even though it divides their product 6. In algebraic number theory 2 is called irreducible in Z [ − 5 ] (only divisible by itself or a unit) but not prime in Z [ − 5 ] (if it divides a product it must divide one of the factors). The mention of Z [ − 5 ] is required because 2 is prime and irreducible in Z . Using these definitions it can be proven that in any ring a prime must be irreducible. Euclid's classical lemma can be rephrased as "in the ring of integers Z every irreducible is prime". This is also true in Z [ i ] and Z [ ω ] , but not in Z [ − 5 ] .

The rings in which factorization into irreducibles is essentially unique are called unique factorization domains. Important examples are polynomial rings over the integers or over a field, Euclidean domains and principal ideal domains.

In 1843 Kummer introduced the concept of ideal number, which was developed further by Dedekind (1876) into the modern theory of ideals, special subsets of rings. Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.

There is a version of unique factorization for ordinals, though it requires some additional conditions to ensure uniqueness.