Prime Conjectures
Frank Kane (Frank.Kane@onslow.school.nz)
Strand: Number and Algebra
Level: 5 (4 – 6)
Aims: Reinforcement of knowledge of prime numbers, development of thinking skills
(conjecture, justify, verify), appreciation of the idea of a proof in mathematics.
Key competencies: Thinking skills, using symbols
Introduction:
Prime numbers have been called the ‘atoms’ of mathematics. These are the indivisible numbers.
Every counting number has a unique expression as the product of prime numbers.
E.g. 70 = 2 × 5 × 7.
Euclid (a Greek mathematician who lived in about the 3th century BC) proved that there are
infinitely many primes, but there are many things about prime numbers that are still a mystery
(chiefly: is there a pattern to their occurrence?), or that seem to be true but have not yet been
proven. These provide a fertile ground for student investigation.
This activity is based on Goldbach’s Conjecture (but students can be told about that after they
have discovered it for themselves):
Every even integer greater than 2 can be written as the sum of two primes.
(Actually, when Christian Goldbach wrote to Leonard Euler in 1742, his conjecture was that
every integer greater than two can be written as the sum of three primes,
but he took 1 to be a prime. Euler replied that it followed from the above usual statement of the
conjecture, adding that he regarded it as self evident.)
The activity
It is assumed students already know what a prime number is and have done an activity like the
‘sieve of Eratosthenes’ to find all the primes up to 100 at least.
In pairs, or small groups, students consider the following:
Write 20 as the sum of two prime numbers.
Is there more than one combination of two primes adding to 20?
So, can all numbers be written as the sum of two primes?
(Note: The primes need not be different numbers.)
They will soon discover that many odd numbers cannot be expressed as the sum of two primes.
They must be encouraged to explain clearly why this is so. Which odd numbers can be written as
the sum of two primes? (Answer: odd numbers that are two more than a prime number.)
After the initial feedback session, where there should be agreement that even numbers seem to be
able to be expressed as the sum of two primes, they can continue the investigation.
Is there a pattern to the number of combinations of pairs of primes that add to
give even numbers?
To answer this you should find all the possible combinations for as many even
numbers as possible. Aim to get up to 100.
Consider how you can present your work. A table could be useful.
How could you show your results on a graph?
You need to end your investigation with some conclusions.
Students could use a spreadsheet to generate a graph:
Number of
combinations
Distribution of number of
representations
7
6
5
4
3
2
1
0
0
20
40
60
80
Even number
You will find a graph on the Wikipedia site on the Golbach conjecture of the distribution of
prime representations for even numbers up to 1 million, which looks intriguing.
Final Discussion
“Have we proved that all even numbers can be written as the sum of two primes?” Discuss what
a conjecture is, and what constitutes a proof. “What would it take to disprove it?” Some
historical context at this stage would be good.
Possible inclusion:
“How might we write the result using algebra?” leading to the use of 2n to represent any even
number and 2n + 1 for an odd number. Simple proofs of number results using algebra could then
be attempted. e.g. The sum of two even numbers is even.
Further investigation:
There is a similar result for odd numbers:
Lemoine’s Conjecture: (also not yet proven)
Every odd number greater than 5 can be written as the sum of a prime number
and two times a prime number. (The two primes could be the same)
e.g. 25 = 19 + 2 × 3
Algebraically: For any n > 2, 2n + 1 = p + 2q for primes p and q.
Another yet to be proven conjecture is the Collatz conjecture, which is a nice little exercise in
arithmetic practice:
Take any counting number.
If it is even, divide by two.
If it is odd, multiply by 3 and add 1.
Continue using the rule above on the answers you get.
The sequence generated will always lead to 1.
(which would then just loop 4-2-1 if you continued.)
Warning: Some starting numbers take a while. If you start with 27, it takes 111 steps to reach 1.