The Fundamental Theorem of
Arithmetic
By Jess Barak, Lindsay Mullen, Ashley
Reynolds, and Abby Yinger
The concept of unique factorization stretches right back to Greek arithmetic and yet
it plays an important role in modern commutative ring theory. Basically, unique
factorization consists of two properties: existence and uniqueness. Existence means
that an element is representable as a finite product of irreducibles (primes), and
uniqueness means that this representation is unique in a certain sense (only one
way of representation with the primes). Unique factorization first appeared as a
property of natural numbers. This property is called the Fundamental Theorem of
Arithmetic (FTA). The FTA is stated as follows: Any natural number greater than 1
can be represented as a product of primes in one and only one way (up to the
order). [1 p. 1]
If n is a whole number then n can be factored uniquely into a product of
prime numbers.
n = 𝑝1 𝑒1 ∙ 𝑝2 𝑒2 ∙ … ∙ 𝑝𝑘 𝑒𝑘
In the instance of the Fundamental Theorem of Arithmetic there were multiple
mathematicians that contributed their thoughts and expertise which finally led to a
complete proof . These mathematicians range from different time periods including
Euclid, Euler, Gauss, and a select few others. The actual credit of the first proof was
given to Gauss who we will discuss in more depth later in this presentation. [1 p. 1-8]
Euclid (365 B.C.):
Although the FTA does not appear in the Elements, there are two very significant
propositions, VII.30 (Euclid’s Lemma) and VII.31, which have a close connection with it.
There is a third proposition, IX.14, which is a uniqueness theorem. In fact, the FTA
follows from the propositions VII.30 and VII.31.
VII.30. If p is prime and p | ab, then p | a or p | b. (Euclid’s lemma)
VII.31. Any composite number is measured by some prime number.
Easily, we get the existence (any natural number greater than 1 can be represented as
a product of primes) by VII.31, and the uniqueness (i.e., this representation is unique
up to the order) by VII.30. [1 p. 2] It is easy to deduce the fundamental theorem of
arithmetic from these propositions by Euclid, and there can be little doubt that had he
recognized this result as fundamental he would have proved it. Euclid, however, was
more interested in being able to list (with proof) all of the divisors of certain integers.
[2 p. 4]
Kamāl al-Dīn al-Fārisī (1200-1300s):
Kamāl al-Dīn al-Fārisī, who died ca. 1320, was a great Persian mathematician,
physicist, and astronomer. His work represents perhaps the most significant step
toward the FTA made by mathematicians before Gauss. One could say that Euclid
takes the first step on the way to the existence of prime factorization, and al-F¯aris¯ı
takes the final step by actually proving the existence of a finite prime factorization in
his first proposition.
PROPOSITION 1. Each composite number can be decomposed into a finite
number of prime factors of which it is the product.
He stated and proved the existence part of the FTA, but he did not state and did not
intend to prove the uniqueness of prime factorization since the FTA was not important
for him. This does not mean he did not know the uniqueness. [1 p. 3]
Jean Prestet (1600s):
Prestet stated neither the existence nor the uniqueness of the FTA. He was influenced by
Euclid and was concerned with divisors. Like Al-Fārisī and Euler he gave the main results
in order to find all the divisors of a given number. In particular his Corollary IX has a
significant role. He provided the corollary,
COROLLARY IX. If the numbers a & b are simple, every divisor (of) aab of the
three a, a, b is one of the three 1, a, aa or one of the different products of these
three by b; that is to say, one of the six 1, a, aa, 1b, ab, aab. Because all the
alternative planes [i.e., obtained by multiplying the different factors two by
two] of the simple a, a, b are aa & ab. [Analogous statements for aabb; aabbb;
aab3cc; aab3ccd].And so with the others.
It is clear that Prestet did not state the FTA in his work because his aim was to make
explicit the relationship between any factorization of a given number into primes and all its
possible divisors. However, Prestet’s results were very close to the FTA, and in the sense of
implying each other his Corollary IX may be considered as equivalent to the uniqueness of
the prime factorization. [1 p. 6]
Euler (1770):
Leonard Euler stated the existence part of the FTA without proving it properly, and
also he gave a statement for the uniqueness part analogous to Al-Fārisī’s
Proposition 9 and Prestet’s Corollary IX.
In Article 41 of Chapter IV of Section I of Part I Euler stated the existence of prime
factorization and provided a partial proof of it. But his proof omits some details.
41. All composite numbers, which may be represented by factors, result
from the prime numbers above mentioned; that is to say, all their factors
are prime numbers. For if we find a factor which is not a prime number, it
may always be decomposed and represented by two or more prime
numbers. When we have represented, for instance, the number 30 by 5×6,
it is evident that 6 not being a prime number, but being produced by
2×3, we might have represented 30 by 5×2×3, or by 2×3×5; that is to
say, by factors which are all prime numbers.
We observe that Euler was only interested in finding all divisors of a number and
he was following the tradition of Al-Fārisī and Prestet. [1 p. 6]
The mathematician we are going to focus on is Carl Friedrich Gauss.
Carl Friedrich Gauss was born in Brunswick, Germany on April 30th 1777. His education
began in 1788 at the Gymnasium where he learned High German and Latin. In 1792 he
entered Brunswick Collegium Carolinum. This was made possible by a scholarship
given to him by the Duke of Brunswick. Three years later, in 1795, he left Brunswick
and went to study at Göttingen University. He was not popular amongst his peers and
was only known to have one friend at this University. Finally in 1807 he became a
professor for astronomy at this same university where he remained until his death in
1855.
As for his discoveries, his first important discovery was the construction of the 17-gon
which he did with a circle and a ruler. He also discovered Bode’s law, the binomial
theorem, the arithmetic-geometric mean, the law of quadratic reciprocity, and the
Fundamental Theorem of Arithmetic. [3] It is here that we focus on his part in
proving of the Fundamental Theorem of Arithmetic. Although many others had played
a part in proving this theorem, it is Gauss who is accredited with proving the theorem.
In 1801, in Article 16 of his Disquisitiones Arithmeticae, was able to clearly state and prove
the uniqueness of the Fundamental Theorem of Arithmetic. Gauss did so with the help of
Euclid and his work within Euclid’s Elements. He even took one of Euclid’s lemma (as
previously stated in the earlier slide about Euclid) and reproduced it as
If neither a or b can be divided by a prime number p, the product ab cannot be
divided by p.
In the proof, it is Euclid’s lemma that really comes into play when proving the uniqueness
aspect of this theorem. Gauss’s proof really focused on the uniqueness. In fact, Gauss
didn’t actually spell out the proof of the existence part of the FTA because he felt this part
was clear from “elementary considerations.” In his work, he states, “It is clear from
elementary considerations that any composite number can be resolved into prime factors,
but it is tacitly supposed and generally without proof that this cannot be done in many
various ways.” [3]
So while many others may receive recognition for their work contributing to this theorem
and its proof, it is Gauss that has been given the credit of the first clear statement and
proof of the Fundamental Theorem of Arithmetic.
[1] Agargun, A. Goksel & Ozkan, E. Mehmet. A Historical Survey of the Fundamental
Theorem of Arithmetic. Turkey: 2001.
[2] Granville, Andrew. The Fundamental Theorem of Arithmetic. Canada: Unknown year.
[3] Website: Johann Carl Friedrich Gauss. http://www-history.mcs.stand.ac.uk/Biographies/Gauss.html (12/1996)
Factoring: the act or process of separating an equation, formula,
cryptogram, etc., into its component parts. [dictionary.com]
In other words, factoring is breaking down a number (n) into its
components or prime numbers (the numbers that when multiplied
together give you n).
Example:
Theorem 1.8 (Fundamental Theorem of Arithmetic) Let n be an integer such
that n > 1. Then
n = 𝑝1 𝑝2 · · ·𝑝𝑘 ,
where 𝑝1 , . . . , 𝑝𝑘 are primes (not necessarily distinct). Furthermore, this
factorization is unique; that is, if
n = 𝑞1 𝑞2 · · · 𝑞𝑐 ,
then k = c and the 𝑞𝑖 ’s are just the 𝑝𝑖 ’s rearranged.
Proof. Uniqueness. To show uniqueness we will use induction on n. The
theorem is certainly true for n=2 since in this case n is prime. Now assume
that the result holds for all integers m such that 2 ≤ m < n, and
n = 𝑝1 𝑝2 · · · 𝑝𝑘 and n= 𝑞1 𝑞2 · · · 𝑞𝑐 so that n= 𝑝1 𝑝2 · · · 𝑝𝑘 = 𝑞1 𝑞2 · · · 𝑞𝑐 ,
Where 𝑝1 ≤ 𝑝2 · · · ≤ 𝑝𝑘 and 𝑞1 ≤ 𝑞2 · · ·≤ 𝑞𝑐 And where p’s and q’s are
primes. By Lemma 1.6 (Euclid’s lemma stating “Let a and b be integers and p
be a prime number. If p|ab, then either p|a or p|b”), 𝑝1 | 𝑞𝑖 for some i = 1, . .
. , c and 𝑞1 | 𝑝𝑗 for some j = 1, . . . , k. Since all of the 𝑝𝑗 ’s and 𝑞𝑖 ’s are prime,
𝑝1 = 𝑞𝑖 and 𝑞1 = 𝑝𝑗 . Hence, 𝑝1 = 𝑞1 since
𝑝1 ≤ 𝑝𝑗 = 𝑞1 ≤ 𝑞𝑖 = 𝑝1 (can be written as 𝑝1 ≤ 𝑞1 ≤ 𝑝1 , so 𝑞1 must be equal
to 𝑝1 since 𝑝1 and 𝑝1 are the same).
By the induction hypothesis,
n’ = 𝑝2 · · · 𝑝𝑘 = 𝑞2 · · · 𝑞𝑐 has a unique factorization since,
n = 𝑝1 𝑝2 · · · 𝑝𝑘 = 𝑞1 𝑞2 · · · 𝑞𝑐
n = 𝑝1 𝑝2 · · · 𝑝𝑘 =𝑝1 𝑞2 · · · 𝑞𝑐 (by substitution)
then, n’ = 𝑝2 · · · 𝑝𝑘 = 𝑞2 · · · 𝑞𝑐
Hence, k = c and 𝑞𝑖 = 𝑝𝑖 for i = 1, ...,k. Keep going and at each stage you’ll pair up a
𝑝𝑗 and a 𝑞𝑖 due to the proceeding argument. You can’t wind up with any primes left
over at the end or else you’d have a product of primes equal to 1 (i.e. 1≠ 5 ∙ 7). So
everything must have paired up and the original factorizations were the same
(except possibly for the order of the factors).∎
In English!.....
2 × 3 × 7 = 42
No other prime numbers would work!
You could try 2 × 3 × 5, or 5 × 11, and none of them will work:
Only 2, 3 and 7 make 42
Now for the existence portion………
Existence. To show existence, suppose that there is some integer
that cannot be written as the product of primes. Let S be the set of
all such numbers. By the Principle of Well-Ordering (“Every nonempty set of positive integers contains a smallest element”), S has a
smallest number, say 𝛼. If the only positive factors of 𝛼 are 𝛼 and 1,
then 𝛼 is prime, which is a contradiction. Hence, 𝛼 is composite and
𝛼 = 𝛼1 𝛼2 where
1 < 𝛼1 < 𝛼 and 1 <𝛼2 < 𝛼. Neither 𝛼1 ∈ S nor 𝛼2 ∈ S, since 𝛼 is the
smallest element in S. So this tells us that 𝛼 can now be written as a
product of primes such that
𝛼1 = 𝑝1 · · · 𝑝𝑟 𝑎𝑛𝑑 𝛼2 = 𝑞1 · · ·𝑞𝑠 , where p and q are prime numbers
Therefore,
𝛼 = 𝛼1 𝛼2 = 𝑝1 · · · 𝑝𝑟 𝑞1 · · ·𝑞𝑠
So 𝛼 ∈ S, which is a contradiction. ∎ [4 p. 39]
[4] Judson, Thomas W. Abstract Algebra: Theory and
Applications. Harvard University: 2008.
Problem: Find the unique prime factorization using the
Fundamental Theorem of Arithmetic. Construct a factor tree and
write out the product of primes.
420
180