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Section 3.2 Prime Numbers

Like atoms making up molecules, prime numbers are the fundamental building blocks of the integers. In this section we see this in The Fundamental Theorem of Arithmetic, and explore some interesting statements prime numbers. We conclude the section with primality testing -- how to determine if a number is prime.

Subsection

Definition 3.2.1. Prime Numbers.

A positive integer \(p\) is called prime if the only positive factors of \(p\) are \(1 \and p\) itself.
A positive integer which is not prime is called \(composite\)
The next logical question students ask is “what about 1?” because its only positive factors are 1 and itself. One is special.

Note 3.2.2. 1 is the loneliest number.

1 is neither prime nor composite. It’s called a unit.
Our next proposition, along with the proof that the square root of 2 is irrational, is the most classic example of a proof by contradiction. It’s hard (impossible?) to prove directly that there are infinitely many primes. Instead, we assume there are finitely many and then find a contradiction.

Proof.

Assume, to the contrary, that there are only finitely many primes. Let’s call them \(p_1, p_2, p_3, \dots p_n\text{.}\) Consider the number
\begin{equation*} N = p_1 \cdot p_2\cdot p_3 \cdots p_n + 1 \end{equation*}
the product of every prime, plus one.
Because \(N\) is larger than the largest prime, \(p_n\text{,}\) we know that \(N\) isn’t in our list of primes and therefore is composite. Then there is some prime \(p_i\) that divides it.
But since \(p_i \divides N\) and \(p_i \divides p_1\cdot p_2\cdot p_3 \cdots p_n\text{,}\) we see that
\begin{equation*} p_i \divides (N - p_1 \cdot p_2\cdot p_3 \cdots p_n) \end{equation*}
meaning that
\begin{equation*} p_i \divides 1 \end{equation*}
But this is a contradiction since the only divisor of 1 is itself. Our assumption was incorrect, and therefore we conclude that there are infinitely many primes.
Over our school careers, we’ve been factoring numbers for a long time. You’ve probably noticed there is “a correct answer” when you produce a factorization. For example, we might write: \(15 = 3\cdot 5\) or \(15 = 5\cdot 3\text{,}\) these are the same factorization, we just changed the order we multiplied. The next theorem, which underlies all arithmetic, tells us we can factor every integer and that there is really a correct answer.

Aside
The Fundamental Theorem of Arithmetic (FTA) is a non-constructive statement. It tells us that we can write every positive integer as a unique product of primes, but it doesn’t tell us how. Instead we need to use trial division and guess-and-check methods.

Example 3.2.5.

Find the prime factorization of the following:
  1. 124
  2. 81
  3. 941
  4. 3780
Video / Answer.
Solution.
Here are factor trees for each of the examples. To construct these, we divide each number by some prime factor and successively divide the remaining factor further until we’re left with only prime factors.
  1. Factor tree for 124
  2. Factor tree for 81
  3. Factor tree for 941, which is prime
  4. Factor tree for 3780
Trying to determine the prime factors of an integer can be really time consuming, especially if a number is prime. Fortunately, the next theorem tells us just how far we have to check in order to test primality or find a prime divisor.

Example 3.2.7. Brute Force Trial Division.

To determine if a number \(n\) is prime, one only needs to test all possible prime divisors up to \(\sqrt{n}\text{.}\)
Use trial division to determine that 251 is prime.
Video / Answer.
Solution.
Since \(\sqrt{251} \approx 15.81\text{,}\) we only need to check prime numbers which are less than 15. These are \(2, 3, 5, 7, 11, 13\) Since none of these divide into 251 evenly, we conclude that 251 is prime.

Example 3.2.8.

Use trial division to determine if any of the following are prime:
  1. 127
  2. 143
  3. 54323
Solution.
  1. Since, \(\sqrt{127} \approx 11\text{,}\) we need only test primes up through 11. Noticing that none of 2, 3, 5, 7, or 11 divide evenly, we conclude 127 is prime.
  2. Here, \(\sqrt{143} \approx 12\) so again we test only up through 11, the largest prime under 12.. We check that 2, 3, 5, 7 do not work but \(143 = 11\cdot 13\text{,}\) so we see that \(143\) is not a prime number.
  3. For \(\sqrt{54323} \approx 233\text{,}\) meaning we need to check every prime less than 233. In order to solve this, let’s find a method to develop that list...

Example 3.2.9. Sieve of Eratosthenes.

In order to generate a list of all prime numbers less than or equal to \(n\text{,}\) start with a tabular listing of integers from 1 to \(n\) we run the following sieve or filter:
  1. Cross off the number 1.
  2. Continuing until you reach \(\sqrt{n}\text{:}\)
    1. Circle the next, non-crossed out number. It’s prime.
    2. Cross out every multiple of this number.
    3. Repeat.
Run the Sieve on this table of integers from 1 to 100.
Table 3.2.10. Integers from 1 to 100
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Video / Answer.
Solution.
Here is the completed table after running the sieve algorithm:
Table of numbers from 1 to 100. Every number is crossed off except for those which are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 83, 89, 97, which is the complete set of primes less than 100.

Subsection Computer Corner - Implementing the Sieve

Implementing the step algorithm is very straight-forward.
  • Make a large list of integers
  • Delete 1
  • Delete every multiple of each prime we encounter
I’m choosing to implement this in Python in an inefficient, but readable way, so that we can run it in a Sage cell below. If you want to see it in your favorite language, you can find it implemented in many, many different languages on Rosetta Code

Aside
We can now return to determining if 54323 is prime. We now have a way of generating a list of primes up to \(\sqrt{54323}\text{,}\) and we can test primality by trial division. We put it all together in the following function is_it_prime:

Exercises Exercises

1.

Find the prime factorization of each of the following integers:
  1. 39
  2. 125
  3. 101
  4. 143
  5. 289
  6. 899
Solution.
  1. \(\displaystyle 39 = 3\cdot 13\)
  2. \(\displaystyle 125 = 5^3 \)
  3. \(101\) is prime - trial divide by 2, 3, 5, 7
  4. \(\displaystyle 143 = 11\cdot 13 \)
  5. \(\displaystyle 289 = 17^2\)
  6. \(\displaystyle 899 = 29 \cdot 31\)

2.

Find the prime factorization of \(10!\) (10 factorial, not “yay ten!”)
Solution.
Recall \(10! = 10 \cdot 9 \cdot 8 \cdot 7 \cdot \dots \cdot 2 \cdot 1\text{.}\) So we can do the following:
\begin{align*} 10! \amp = 10\cdot 9 \cdot 8 \cdot 7\cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 \\ \amp = (2\cdot 5)\cdot 3^2 \cdot 2^3 \cdot 7\cdot (2\cdot 3) \cdot 5 \cdot 2^2 \ cdot 3 \cdot 2 \cdot 1 \\ \amp = 7^1 \cdot 5^2 \cdot 3^4 \cdot 2^8 \end{align*}

3.

Using the is_it_prime function above (or write your own in your favorite language!), factor the following large integers.
  1. 654321
  2. 123456
  3. 987654321
  4. 123456789
Hint.
Begin by changing integer_to_check to 654321 and see that 3 divides it. Then make integer_to_check = 218107 and repeat...
Solution.
  1. \(\displaystyle 654321 = 3\cdot 218107 \)
  2. \(\displaystyle 123456 = 2^6 \cdot 3 \cdot 643 \)
  3. \(\displaystyle 987654321 = 3^2 \cdot 17^2 \cdot 379721\)
  4. \(\displaystyle 123456789 = 3^2 \cdot 3607 \cdot 3803\)

4.

Prove that every prime number greater than 3 is either one more or one less than a multiple of 6.
Hint.
Prove the contrapositive by cases.
Solution.
This question is of the form “if \(n\) is a prime number greater than three, then either \(n=6k+1\) or \(n=6k-1\) for some integer \(k\)”.

Proof.

(by contraposition). Assume that \(n\) is not a number of the form \(n=6k+1\) or \(n=6k-1\) for some integer \(k\text{.}\) This gives us four total cases:
Case 1: Assume \(n=6k\) for some integer \(k\text{.}\) Then \(n\) is not a prime number, since \(n\) is divisible by both 2 and 3.
Case 2: Assume \(n = 6k+2 = 2(3k+1)\) for some integer \(k\text{.}\) Then \(n\) is divisible by 2 and is hence not prime.
Case 3: assume \(n = 6k+3 = 3(2k+1)\) for some integer \(k\text{.}\) then \(n\) is divisible by 3 and is hence not prime.
Case 4: assume \(n = 6k+4 = 2(3k+2)\) for some integer \(k\text{.}\) then \(n\) is divisible by 2 and is hence not prime.
Thus if \(n\) is not of the form \(n=6k+1\) or \(n=6k-1\) for some integer \(k\text{,}\) \(n\) is not prime. By contraposition, if \(n\) is a prime greater than three then \(n\) is of the form either \(n=6k+1\) or \(n=6k-1\) for some integer \(k\text{.}\)

5.

Give a proof for Theorem 3.2.6
Hint.
What happens if we assume that every prime divisor of \(n\) is bigger than \(\sqrt{n}\text{?}\)
Solution.
Our proof will go like this: factor \(n\) into three parts, \(p_1, p_2, \and b\) where \(p_1 \and p_2\) are prime and \(b\) is whatever is left with the factorization, such as \(10 = 2\cdot5\cdot1\) where \(b=1\) or \(60=2\cdot5\cdot6\) where \(b=6\text{,}\) but assuming those prime factors are bigger than \(\sqrt{n}\text{,}\) and look for the contradiction.

Proof.

Assume, to the contrary, that every prime divisor of \(n\) is greater than \(\sqrt{n}\text{.}\) Since \(n\) is composite, it has at least two prime factors, \(p_1, p_2\) so that \(n = p_1 p_2 b\) where \(b\) is some positive integer. But since \(p_1 \and p_2\) are a prime divisors of \(n\text{,}\) by our assumption, \(p_1 \gt \sqrt{n} \and p_2 \gt \sqrt{n}\text{.}\) So:
\begin{equation*} n = p_1 p_2 b \gt \sqrt{n} \cdot \sqrt{n} \cdot b = nb \end{equation*}
But since \(n\gt 1 \) and \(b\in \Z^+\text{,}\) \(n\gt nb\) is a contradiction. This means our original assumption, that every prime factor of \(n\) was greater than \(\sqrt{n}\) was wrong, and therefore we conclude that \(n\) has a prime factor less than \(\sqrt{n}\text{.}\)
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