This is a very brief return to the topic of binary fractional representations of the inverse of primes. I had written a bit about this previously here:
Binary representation of inverted primes
and followed up here:
Binary representation of inverted primes, continued…
In Chapter 14 (“Primitive Roots”) of Manfred Schroeder’s other book, Number Theory in Science and Communication, he discusses a number of important number theoretic concepts of relevance to cryptography, including the notions of primitive roots, index, period, and order. Each of these concepts has relevance in modular arithmetic.
To get to the point, a fraction will have a terminating decimal representation if and only if the only prime factors of the denominator are 2 and/or 5, the only prime factors of 10. Denominators of other rational fractions with other factors will yield repeating decimal fractions, and irrational numbers will yield non-repeating decimal fractions.
Similarly, with binary fractional representation, only fractions with denominators that are a power of 2 (the only prime factor of 2) will yield terminating binary fractions, other rational fractions will yield repeating fractions, and irrational numbers non-repeating.
The point I want to get to, which relates to prior posts, is about the LENGTH of the repeating portion for the repeating binary fractions of inverted PRIME NUMBERS. Schroeder makes the point that this length will be equal to the ORDER of 10 (for decimal representation) or 2 (for binary representation) MODULO the denominator. If 10 (or 2 for binary representation) is a PRIMITIVE ROOT of the PRIME denominator, p, then the length of the repeating segment will be equal to (p – 1).
What I showed previously was that for many inverted primes, the repeating length was equal to (p – 1). In most other cases, it was (p – 1)/2 (although for p = 31 it is 5, or (p – 1)/6), and in one special case (1/2) it is non-repeating (as expected, since 2 divides 2).
We should therefore be able to conclude that 2 is a PRIMITIVE ROOT of those primes where the repeating length is (p – 1), and is not a PRIMITIVE ROOT for the other primes. Note that all prime numbers, integer powers of prime numbers, and integer powers of prime numbers times 2, all have PRIMITIVE ROOTS (not all integers have them), so these other prime numbers must have different PRIMITIVE ROOTS.
Of the numbers I examined (2 to 67), it would seem the following DO NOT have 2 as a PRIMITIVE ROOT:
7, 17, 23, 31, 41, and 47.
All the other, (13 other primes including 2) have 2 as a PRIMITIVE ROOT.
I wonder whether this has any particular significance, or is rather just some random, irrelevant fact about these numbers. The other interesting (if only to me) aspect of the binary fraction representations of inverted primes with 2 as a PRIMITIVE ROOT is that they each have exactly the same number of zeroes and ones, whereas the inverted primes without 2 as a primitive root have some other fraction (which is sometimes 1/2) of 1s among the repeating terms.
More to follow, but for now, back to work-related research efforts…