“I have to show this magic number.” Justin grabbed his calculator and typed a number on its tiny screen then passed it to me.

“142857. What’s so special about it?” I asked.

“Multiply by 2 and see what happens.”

“285714. Oh, it’s the same digits as the original but shifted two places to the left. That’s cool.”

“Now add the original number to your answer.”

“OK, and the result is 428571. So it’s moved the digits one place to the right this time.”

“And it keeps shifting the same digits around until you reach 7 times the magic number.”

“Let’s see. 7 times 142857 is 999999. Hey, why does it do that?”

“I don’t know, but if you keep going past this some other interesting numbers develop.”

Later I did have a look at these, you can too. What patterns can you see? I thought about the magic number for a while and wondered whether there might be any others. How would you attempt to find some more?

I decided to try to find a number that when doubled, shifted its digits exactly one place to the left. Such a number, I reasoned, would need to follow a certain pattern. Each digit would have to have a value that was equal to twice that of the digit to its right. You can construct this number with me. Start by writing the digit 2 on the right side of a piece of paper. Double it to get 4, which is the next digit. Twice 4 gives you 8, doubling 8 gives 16. Write down the 6 and carry the 1. Twice 6 is 12 plus 1 is 13. Write down the 3 and carry the 1. So far you should have the number 36842. Continue this process until the pattern of digits starts to repeat. You will now have an 18 digit number.

This is the only number I know that shifts its digits in this manner when doubled. An amazing thing is that it keeps doing this if you continue to add the original number to your answer. How many times can you do this before the pattern breaks down?

Intrigued by these numbers I contacted the Mathematics Faculty of my local university to find out more. What I discovered was of prime importance. The magic numbers are related to prime numbers. The first is part of the decimal expansion of the fraction 1/7 and the second is part of 2/19. Use a calculator or long division to see this. For the prime fraction 2/19, you will need a calculator or a spreadsheet program that can handle large numbers.

Both 7 and 19 are prime numbers, which means that they have no factors except themselves and 1. In other words there no other numbers that divide them exactly. The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, 19. Composite numbers such as 12 have several factors: 1, 2, 3, 4, 6, 12. How can we find more primes?

Draw a table with 10 columns and 10 rows, and write in the numbers from 1 to 100. Cross out the number 1, because that is not considered to be a prime number. Circle the number 2, to show it is a prime number. Cross out any numbers that can be divided by 2, such as 4, 6, 8 . . . all the way to 100. Circle the number 3 and cross out any of its multiples -- 6, 9, 12 . . .The next number that is not crossed out is the 5 so circle it and repeat as before. Continue circling primes and crossing out the others until you reach the end of your table. As you can see there are quite a few prime numbers less than 100. Some of them could have the same magical properties that we discovered before. For example the decimal part of 1/61 has a 60 digit pattern that is similar to 142857. What primes can you find that produce magic numbers?