Canonical Prime Decomposition

Standard Form of an Integer

Unique prime factorization says that every integer $n>1$ can be written as a product of primes. The canonical prime decomposition is the ordered and exponentiated version of this factorization.

For every integer $n>1$ , there exist distinct primes

p_1<p_2<\cdots<p_r

and positive integers

\alpha_1,\alpha_2,\ldots,\alpha_r

such that

n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}.

This expression is canonical because the primes are written in increasing order and equal primes are grouped into powers.

For example,

840=2^3\cdot3\cdot5\cdot7.

No other ordered list of primes and exponents gives the same integer.

Exponents as Arithmetic Data

The exponent $\alpha_i$ records how many times the prime $p_i$ occurs in the factorization of $n$ .

For example,

360=2^3\cdot3^2\cdot5.

This means

360=2\cdot2\cdot2\cdot3\cdot3\cdot5.

The prime $2$ appears three times, the prime $3$ appears twice, and the prime $5$ appears once.

Thus canonical decomposition converts multiplication into a finite record of prime exponents.

Prime Valuations

For a prime $p$ , the exponent of $p$ in the prime factorization of $n$ is called the $p$ -adic valuation of $n$ . It is denoted by

v_p(n).

Thus

v_p(n)=\alpha

if $p^\alpha\mid n$ but

p^{\alpha+1}\nmid n.

For example, since

360=2^3\cdot3^2\cdot5,

we have

v_2(360)=3, \qquad v_3(360)=2, \qquad v_5(360)=1.

If a prime $p$ does not divide $n$ , then

v_p(n)=0.

Reconstructing the Integer

The collection of all valuations determines the integer completely:

n=\prod_p p^{v_p(n)}.

Although the product is written over all primes, only finitely many exponents are nonzero. Therefore the product is finite in substance.

For example,

72=2^3\cdot3^2

means

v_2(72)=3, \qquad v_3(72)=2,

and

v_p(72)=0

for every other prime $p$ .

The canonical decomposition may therefore be viewed as a coordinate system for positive integers, with one coordinate for each prime.

Multiplication in Canonical Form

Canonical decomposition makes multiplication simple.

a=\prod_p p^{\alpha_p}

and

b=\prod_p p^{\beta_p},

then

ab=\prod_p p^{\alpha_p+\beta_p}.

Multiplication of integers becomes addition of prime exponents.

For example,

12=2^2\cdot3

and

45=3^2\cdot5.

Therefore

12\cdot45=2^2\cdot3^3\cdot5=540.

The exponent of each prime in the product is the sum of its exponents in the factors.

Divisibility in Canonical Form

Divisibility also becomes transparent.

a=\prod_p p^{\alpha_p}

and

b=\prod_p p^{\beta_p},

then

a\mid b

if and only if

\alpha_p\le \beta_p

for every prime $p$ .

For example,

18=2\cdot3^2

and

252=2^2\cdot3^2\cdot7.

Since every exponent in $18$ is less than or equal to the corresponding exponent in $252$ , we have

18\mid252.

But

20=2^2\cdot5

does not divide $252$ , because $5$ appears in $20$ but not in $252$ .

GCD and LCM

The canonical decomposition gives direct formulas for greatest common divisors and least common multiples.

a=\prod_p p^{\alpha_p}

and

b=\prod_p p^{\beta_p},

then

\gcd(a,b)=\prod_p p^{\min(\alpha_p,\beta_p)}

and

\operatorname{lcm}(a,b)=\prod_p p^{\max(\alpha_p,\beta_p)}.

For example,

72=2^3\cdot3^2

and

120=2^3\cdot3\cdot5.

Hence

\gcd(72,120)=2^3\cdot3=24

and

\operatorname{lcm}(72,120)=2^3\cdot3^2\cdot5=360.

The gcd uses the smaller exponent of each prime. The lcm uses the larger exponent.

Powers and Perfect Powers

Canonical decomposition also characterizes perfect powers.

An integer $n>1$ is a perfect square if

n=m^2

for some integer $m$ . In prime decomposition, this happens exactly when every exponent $v_p(n)$ is even.

For example,

144=2^4\cdot3^2

is a square because all exponents are even:

144=12^2.

More generally, $n$ is a perfect $k$ -th power if every exponent in its canonical decomposition is divisible by $k$ .

Role in Number Theory

Canonical prime decomposition is one of the main languages of elementary number theory. It converts multiplicative questions about integers into questions about exponents.

Divisibility becomes coordinatewise comparison. Multiplication becomes coordinatewise addition. Greatest common divisors and least common multiples become coordinatewise minimum and maximum.

This viewpoint is simple but powerful. It prepares the way for arithmetic functions, valuations, local methods, and the broader idea that global arithmetic structure can be studied prime by prime.