Amazing Number Facts No

No 33

One over Primes

1/2 0.5 Terminates 1/3 0.33333... Repeating block: 1 digit 1/5 0.2 Terminates 1/7 0.1428571428... Repeating block: 6 digits 1/11 0.090909... Repeating block: 2 digits 1/13 0.0769230769... Repeating block: 6 digits 1/17 0.05882352941176470588... Repeating block: 16 digits 1/19 0.0526315789473684210526... Repeating block: 18 digits 1/23 0.04347826086956521739130434... Repeating block: 22 digits Notice that for some primes the size of the repeating block is 1 less than the prime.
These are called Golden Primes or Long Primes.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
9 out of the 25 primes less than 100 are golden, that's 36%.
Emil Artin made a conjecture that in the long run the proportion of primes that are golden is given by:

Can you spot the pattern in the terms of this product?
You might like to calculate the first few terms of this product and compare it with 36%.
What are the next few golden primes? If you have answers, discoveries, new questions etc to do with this Number Fact then...
E-mail us at madrascollege.maths@fife.gov.uk
...and we will publish them here.