Skip to 0 minutes and 7 secondsMICHAEL ANDERSON: So let's have a look at finding some prime numbers. One mathematician who came up with one method was Aristophanes, an ancient Greek mathematician. Now his method is called the Sieve of Aristophanes. So should we have a look to see how it works?

Skip to 0 minutes and 22 secondsPAULA KELLY: OK, so I can see here we have 100 grid, so 1 to 100, however this number 1 is missing. Why is this?

Skip to 0 minutes and 30 secondsMICHAEL ANDERSON: Well prime numbers are defined as having just two factors, 1 and itself. So the number 1 only has one factor so it doesn't count as prime number. The smallest number is 2.

Skip to 0 minutes and 41 secondsPAULA KELLY: Because 2 has two different distinct factors, 1 and itself. OK, so we'll circle that. So 2 is our first prime number, OK.

Skip to 0 minutes and 50 secondsMICHAEL ANDERSON: So any number that's in the 2 times table is also divisible by 2. So automatically, they're not going to be prime.

Skip to 0 minutes and 57 secondsPAULA KELLY: OK, so you can get rid of 4, we can get me to 6, 8, 10. And we can see a pattern here. I think that ends in a-- we've got 2 as our prime number. It ends in a 2, 4, 6, 8, or a 0 is not prime. OK.

Skip to 1 minute and 14 secondsMICHAEL ANDERSON: So just like Aristophanes, we're almost sifting out the numbers that are going leave us with primes. So all the numbers in the two times tables we can get rid of.

Skip to 1 minute and 27 secondsPAULA KELLY: OK, so next you have three. The factors or 3 are 1 in 3, so two factors, 1 and itself.

Skip to 1 minute and 34 secondsMICHAEL ANDERSON: Yeah, so once you've finished the colouring in process, the next largest number, in this case 3, is our next prime.

Skip to 1 minute and 41 secondsPAULA KELLY: OK, so we'll keep 3. So with that same reasoning with our sieve, we can get rid of the rest the multiples of 3.

Skip to 1 minute and 48 secondsMICHAEL ANDERSON: Yep. So 6 is already gone, but we're going to have to discount 9.

Skip to 1 minute and 54 secondsPAULA KELLY: 12 is gone? So we need 15, 18 has gone, 21, 24 is gone. 27, 30, and so on.

Skip to 2 minutes and 3 secondsMICHAEL ANDERSON: And it can really help us with this to notice some patterns, so all the 3 times tables seem to be in these diagonals going down. So we can go at the 6, 15, 24, and then 51 is going to be out.

Skip to 2 minutes and 14 secondsPAULA KELLY: OK. And then our next diagonal, we'll start from 9. So we go across, all these can go as well. These are multiples of 3. And our next diagonal starts from 30, so 39.

Skip to 2 minutes and 31 secondsAnd our next one starts from--

Skip to 2 minutes and 33 secondsMICHAEL ANDERSON: 60.

Skip to 2 minutes and 36 secondsPAULA KELLY: And finally--

Skip to 2 minutes and 37 secondsMICHAEL ANDERSON: 90 and 99.

Skip to 2 minutes and 39 secondsSPEAKER 2: OK.

Skip to 2 minutes and 39 secondsMICHAEL ANDERSON: So you can see how this is going to be quite a time consuming process to find these primes. So the next prime number would be five, and we can keep on going.

Skip to 2 minutes and 48 secondsPAULA KELLY: OK, so again, we know multiples of 5 en in a 5 or a 0. All 0 are gone, also in multiples of 2. We'll get rid of all our 5's.

Skip to 3 minutes and 2 secondsMICHAEL ANDERSON: And the next number we've got left is 7, so seven is the next prime number.

Skip to 3 minutes and 7 secondsPAULA KELLY: OK, because factors of seven are just 1 and 7. So we have a few more. 13 is also prime. 17 is prime, just two factors, 1 and 17. 19, 23, 29, 31, and so on.

Skip to 3 minutes and 30 secondsMICHAEL ANDERSON: Yeah. Should we take a look to see what the completed grid would look like?

Skip to 3 minutes and 33 secondsPAULA KELLY: OK. So we have here our completed grid. So we've used our sieve to find all our prime numbers under 100.

Skip to 3 minutes and 42 secondsMICHAEL ANDERSON: Yep, and you can see them all there. And just looking at them, there doesn't really seem to be much of a pattern. It's really hard to predict when the next prime number might appear. They're in lots of different columns, there's different gaps, different spaces between them. It's really hard to tell where a prime number may just pop up.

Skip to 3 minutes and 58 secondsPAULA KELLY: OK, so let's maybe explore it further.

Skip to 4 minutes and 2 secondsMICHAEL ANDERSON: So let's use a different grid to see if we can find any kind of pattern with the prime numbers. So this is quite an unusual grid. It's got that one just in a box by itself there in the top left, and then we go from 2 to 7, and then we repeat, so 8 to 13 going up by 6 each time.

Skip to 4 minutes and 21 secondsPAULA KELLY: OK, so if we're going to use this to find some patterns of prime numbers, should we circle our primes?

Skip to 4 minutes and 26 secondsMICHAEL ANDERSON: Mm-hmm.

Skip to 4 minutes and 27 secondsPAULA KELLY: OK. Should I start by circling 1?

Skip to 4 minutes and 29 secondsMICHAEL ANDERSON: No, definitely not. So 1 isn't a prime number. It's only got one factor. The smallest prime number and the only even prime number is 2.

Skip to 4 minutes and 37 secondsPAULA KELLY: OK so we'll circle 2.

Skip to 4 minutes and 40 secondsMICHAEL ANDERSON: And so the next number is 3, and then 5, and then 7. So we'll carry on circling them and see if any patterns kind of develop. So we've got 11, 13, 17, and 19, 23, 29, 31, and 37.

Skip to 5 minutes and 2 secondsPAULA KELLY: OK, so is there a pattern there so far?

Skip to 5 minutes and 8 secondsMICHAEL ANDERSON: It doesn't really look like it too much. Two and three seem to be kind of on their own a little bit, and then every other prime number seems to be down in this column here, and in this column here. Although 25 isn't prime so that seems to break it up a little bit. Should we circle a few more and see if we can see anything else?

Skip to 5 minutes and 26 secondsPAULA KELLY: Yeah, we'll see if that same pattern continues. OK. So can we see a pattern now?

Skip to 5 minutes and 33 secondsMICHAEL ANDERSON: Well just like we kind of noticed before, actually all the primes after 2 and 3 seem to either be in this column or in this column. Which to say that it looked like there was no pattern previously, seems a little bit odd. So so far the pattern we're getting is all of our prime numbers, with the exception of two and three, are one less or one more than a multiple of six.

Skip to 5 minutes and 58 secondsPAULA KELLY: Oh yeah, because this column here, it starts with 6. And then because we're going up by 6 each time, 6, 12, 18, 24, all of these are going to be multiples of 6. So I suppose you're never going to get a prime number in there because they're always going to be divisible by 6.

Skip to 6 minutes and 14 secondsMICHAEL ANDERSON: We're saying that prime numbers, with the exception of 2 and 3, are one less or one more than a multiple of six. But we're saying that not all numbers that are one less or one more than a multiple of six are prime, because we have 35. That's one less than a multiple of 6, it's not prime.

Skip to 6 minutes and 34 secondsPAULA KELLY: No, but at least we've made a start in maybe narrowing down where we can find prime numbers.

Prime numbers

Prime numbers are the building blocks of the integers and fascinate mathematicians. However, not just mathematicians are intrigued by prime numbers. Author Mark Haddon’s character, Christopher, in the book ‘The Curious Incident of the Dog in the Night-Time said:

“I think prime numbers are like life. They are very logical but you could never work out the rules, even if you spent all your time thinking about them.”

Being able to express integers as the product of their prime factors is very useful when finding highest common factors or lowest common multiples, which in itself is useful when performing other mathematical tasks such as adding fractions. Later this week we explore how students may use prime factorisation in other areas of the mathematics curriculum.

There are dozens of important uses for prime numbers. Cicadas time their life cycles by them, modern screens use them to define colour intensities of pixels, and manufacturers use them to get rid of harmonics in their products. However, these uses pale in comparison to the fact that they make up the very basis of modern computational security. Whether it’s communicating your credit card information to online retailers, logging into your bank, or sending a manually encrypted email to a colleague, we are constantly using computer encryption. That means we are constantly using prime numbers, and relying on their odd numerical properties for protection of the digital way of life.

In this video Paula and Michael define a prime number and consider an activity to help find the values of small prime numbers. In addition, Paula and Michael explore how, using a specifically designed grid, it is possible to find some pattern in the sequence of prime numbers

Discussion

We have two discussion questions for you to deepen your understanding of prime numbers.

  1. Paula and Michael define a prime number as a number having just two factors: one and itself. This is a definition found in many text books. Use this definition to decide whether -1 is a prime number. Do you think -1 is a prime number? If -1 is a prime number then all is fine however if -1 is not a prime number how would you change Paula and Michael’s definition?

  2. How would you use the pattern described in the video to convince yourself that the number 740737 could possibly be a prime number and is worth further investigation but the number 740739 is definitely not a prime number

Problem worksheet

Now complete question 7 from this week’s worksheet.

Teaching resource

This collection of resources contains a variety of activities for use in the classroom to explore the properties of factors, multiples and primes. The collection also links to complimentary tasks on the NRICH website.

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This video is from the free online course:

Maths Subject Knowledge: Understanding Numbers

National STEM Learning Centre