Random numbers are in great demand in many applications, from hypothesis testing to cryptography, and random number generators are similarly sought-after. One simple yet excellent source of random numbers is the decimal expansion of reciprocals of the form 1/q, where q is a "suitable" prime number.
In this paper, I use some well-known theorems in number theory to deduce a sufficient condition on q, such that 1/q generates q - 1 decimal digits (the maximum possible before repeating), which can be used to form a stream of random numbers.
Specifically, I show that such q are a sub-species of Sophie Germain primes, S, such that
q = 2S + 1 where S and 2S +1 are prime, and S is of the form 3, 9 or 11 mod 20
Thus "suitable" prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to S = 3, 11, 23, 29, 83, 89, etc.).
For example,
1/23 = 0.0434782608695652173913......(22 places) before repeating again
The largest value of S I am aware of that meets these criteria is
S = (39051 x 2^6002) - 1
a 1812-digit Sophie Germain prime of the form 3 mod 20 discovered by Wilfred Keller (cited by Yates, S in "Tracking Titanics", a contribution to The Lighter Side of Mathematics (Proceedings of the Eugene Strens Memorial Conference on Recreational Mathematics and its History), Guy, R. K., & Woodrow, R.E. eds. published by the Mathematical Association of America Washington DC 1994, p 357). This titanic Sophie Germain prime would generate an equally titanic decimal expansion around 10^1812 digits long before repeating.