## Classroom Report on Ghostly Numbers

November 07, 2016
• ### How did you go with our Ghostly Numbers problem?

It was definitely appreciated up at Riverton Primary School, where groups of students appeared to have really enjoyed their time spent uncovering them. The attached file is a report of their work that uses the STAR model both as a framework for the report and as a guide that the students used as they worked on the problem.

Yes, the Riverton students found that ghostly numbers are all, except for the first one, multiples of 6 (which known to be quite a spooky number for some). I thought I could also give the reason why this is the case, as the argument includes some simple maths that will surely make sense to all readers.

A ghostly number is one that lies between two prime numbers that are larger than 4. These primes are all odd numbers and so the number between them will be even, or, put another way, it will be a multiple of 2. Now take any three consecutive numbers. One of them has to be a multiple of 3, and since the two numbers either side of a ghostly number are primes, neither of them can be a multiple of 3, which means that a ghostly number has to be a multiple of 3. Any number that is a multiple of both 2 and 3 has to be a multiple of 2  × 3 = 6. So there we are - all ghostly numbers apart from the first one, are multiples of 6. Why not the firs one? Well, the first ghostly number lies between the twin primes 3 and 5 and in the run of three consecutive numbers 3, 4 and 5, it is the first one that is a multiple of 3.

I hope you have a moment in which to read the report as it is a great example of the STAR model guiding students to a successful conclusion.