Prime number generalisation

The annoying (or fascinating) thing about prime numbers is that there is no pattern in their occurrence within the counting numbers.

In my last post I set this problem:

Here is a generalisation: "In every third decade starting with 10–19 there are at least three prime numbers."

[Every third decade would be: 10–19, 40–49, 70–79, 101–109, and so on.]
Is this a valid generalisation? Or can you find a counterexample?

First, let's say, without doing any investigation, that I would be very surprised if this proved to be valid, because patterns in the occurrence of prime numbers are elusive. I'm fairly confident that we should find plenty of counterexamples.

You could do this just by checking each decade until you find one that doesn't work. I'll leave you to do that, if you wish. You may have to check quite a lot of numbers! Note that you need to check only the four numbers in each decade that end in 1, 3, 7 and 9. Clearly, the even numbers and those ending in 0 and 5 are not prime. So, for example, in the decade 220–229 you have to check only 221, 223, 227 and 229.

Here's an approach that is more analytical.

The four crucial numbers in each decade we are considering are a multiple of 30 plus 11, or 13, or 17, or 19. For example, in the decade 220–229, the four numbers we need to check are 210 + 11, 210 + 13, 210 + 17 and 210 + 19, where 210 is a multiple of 30.

So, if we choose a multiple of 30 that is also a multiple of, say, 11 and 13, then the first two of the four crucial numbers will definitely not be prime Because they will be multiples of 11 and 13, respectively.

So, the decade I will manufacture as my counterexample is the one that begins with 11 x 13 x 30 + 10 = 4290 + 10. That is: the decade 4300–4309.

The argument is that 4301 = 4290 + 11, which must be a multiple of 11 because 4290 is a multiple of 11. (In fact 4301 = 391 x 11). And 4303 = 4290 + 13, which must be a multiple of 13 because 4290 is a multiple of 13. (In fact 4303 = 331 x 13).

So, there's my counterexample: the decade 4300–4309 does not contain three or more primes, because 4300, 4301, 4302, 4303, 4304, 4305, 4306, 4308 are all definitely not prime.

I do not need to check 4307 or 4309 to make my case. But, as it happens, neither of these is prime: 4307 = 59 x 73 and 4309 = 31 x 139.

So, by chance, I have discovered a decade without any prime numbers!

What is a general statement?

When we finally get rid of the crumbling 1999 National Curriculum for Key Stage 2, we will be able to bid farewell to a really duff example, which occurs in the Using and Applying Number section, under the heading 'Reasoning'.

Statement j is that pupils should be taught to understand and investigate general statements. Fair enough. But then an example is given that makes me wonder if the authors themselves understand general statements. This is it: 'there are four prime numbers less than 10'.

This is a true statement, but it is not a general statement. A general statement (generalization) is an assertion that something is true in general, in other words, in all the cases in a particular set.

An example of a generalization would be: 'there are four prime numbers in every decade'. As it happens this generalization is invalid. Assuming that the decades are 0–9, 10–19, 20–29, and so on, there are four prime number in the first decade (2, 3, 5, 7), four in the second decade (11, 13, 17, 19), but only two in the next (23, 29). So the decade 20–29 provides a counterexample showing that the generalization is false.

But the statement is nevertheless a generalization, because it is an assertion that something is true in a number of cases. General statements in mathematics usually use words such as: all, every, any, each, always, whenever.

Here's a generalization for you to investigate: in every third decade starting with 10–19 there are at least three prime numbers. [Every third decade would be: 10–19, 40–49, 70–79, 101–109, and so on.]

Is this a valid generalisation? Or can you find a counterexample?

The Housekeeper and the Professor: book recommendation

Yesterday morning I was chairing the West Midlands regional meeting for the TDA's Student Associates Scheme at Staffordshire University. I left at about 1.20 pm, and, thanks to a broken down train somewhere around Derby, got home to Norwich at about 8 pm!

However, this gave me the ideal opportunity to finish the novel I was reading: a charming, subtle book that Jenny, one of my two daughters, astutely chose for me at Christmas. So I recommend to you: Yoko Agawa (2009, translated by Stephen Snyder) The Housekeeper and the Professor. London: Harvill Secker.

It's the story of a housekeeper and her son who look after a Japanese professor of mathematics, who has lost all but 80 minutes of short-term memory in an accident. But deeply embedded in his memory is his love of number theory, a world into which the housekeeper and her son are gradually drawn. It's truly enchanting.

To get the most out of it you may need to brush up on multiples, factors, primes and number patterns: Chapters 11 and 12 of the current edition of Mathematics Explained for Primary Teachers!