Even Numbers as the Sum of Two Primes (The Goldbach Conjecture)

Summary

This lesson is about an interesting property of all even numbers greater than 2. Every even number greater than 2 is the sum of two prime numbers.

Background – Prime numbers

A prime number is a number that cannot be formed by multiplying two smaller numbers. For example, 5 is a prime number because only 1 x 5 = 5. 6 is not a prime because you can multiply two smaller numbers, 2 and 3, to make 6. Prime numbers under 100 are:

2            3            5            7            11          13              17          19          23         29         31          37          41          43         47          53             59          61          67          71          73          79          83         89         97         

Numbers, other than 0 and 1, that are not prime are call composite numbers. All composite numbers can be formed by multiplying two smaller numbers.

Examples

Here are examples of how even numbers can be expressed as the sum of two prime numbers:
4 is the sum of 2 + 2

6 is the sum of 3 + 3 (the two numbers do not have to be the same as you’ll see in the next example)

8 is the sum of 3 + 5

10 is the sum of 5 + 5 (10 can also be expressed as the sum of two other primes, 3 + 7)

12 is the sum 5 + 7

 

Exercises

1.  Continue with the number 14. Using the list of prime numbers (above), find how each even number can be expressed as the sum of two prime numbers. Try to complete the exercise for all even numbers up to 40.

2. There will always be at least one way to add two primes to form each even number but in some cases, as shown above for 10, there can be two or more ways to add two primes to form the number. Find other even numbers like 10 for which there are two or more ways to add two primes to form the number.

3.  The number 100 is interesting in that there are six ways in which the sum of two primes equal 100. Can you find all six pairs of prime numbers that add up to 100?

4.  This pattern of adding two prime numbers only works for forming even numbers; however, it sometimes works for odd numbers - but not all. For example: 5 = 2 + 3; 7 = 2 + 5; but 11 cannot be expressed as the sum of two prime numbers. To check, here are all the ways two numbers add up to 11:
1 + 10 (10 is not prime)

      2 + 9 (9 is not prime)

3 + 8 (8 is not prime)

4 + 7 (4 is not prime)

5 + 6 (6 is not prime)

Continue the exercise with higher odd numbers; either find a pair of prime numbers that add up to the number or prove (as done above for 11) that the number cannot be formed by the sum of two prime numbers. 

For parents, teachers and advanced thinkers - How does it work?

This lesson is based on the Goldbach Conjecture which states that all even numbers greater than 2 can be expressed as the sum of two prime numbers.  A conjecture is a mathematical statement that is generally accepted as true but has not been proven. A formally proven statement is called a theorem. While the Goldbach Conjecture has not been proven, it has been confirmed to be true using computers for even integers beyond 1,000,000,000. Monetary prizes are available for the person who can formally prove the Goldbach statement to be true. If a single even number was found that could not be stated as the sum of two prime numbers, then the Goldbach statement would be disproven. However, given the very large numbers that have already been tested, finding such a starting number is unlikely.

For additional information:

Goldbach conjecture info: http://www.mi.sanu.ac.rs/vismath/garavaglia2008/index.html

 

Exercise key and comments:

1.      (Just one pair of prime numbers is listed for each even number; there may be additional pairings that work)
14 = 7 + 7
16= 3 + 13
18= 7 + 11
20= 7 + 13
22 = 3 + 19

24 = 5 + 19
26 = 13 + 13
28 = 5 + 23
30 = 7 + 23
32 = 3 + 29
34 = 17 + 17
36 = 7 + 29

38 = 19 + 19
40 = 3 + 37

 

 

    2. Examples of even numbers formed by multiple pairs of two primes include:
         10 = 3 + 7; 10 = 5 + 5

16 = 5 + 11; 16 = 3 + 13

20 = 3 + 17; 20 = 7 + 13

 

3.  The number 100 can be formed the sum of six different pairs of prime numbers: 

100 = 3 + 97; 100 = 11 + 89; 100 = 17 + 83; 100 = 29 + 71; 100 = 41 + 59; 100 = 47 + 53

    4.    13 = 2 + 11
            15 = 2 + 13
            19 = 2 + 17
            21 = 2 + 19
            25= 2 + 23

Additional counter examples include: 17, 23, 27, 29, 37…

Update 1/18/2022 - a recent post gives some additional insight to this problem: Math Vacation: Additional Thoughts on the Goldbach Conjecture (jamesmacmath.blogspot.com).