I’ve generated the binary representations of a few more inverted primes, from 37 to 67, to investigate whether the observed patterns continue.
In the following, I’ve used superscripts to represent multiple 1’s or 0’s, as illustrated for 1/37:
I’ve noted additional patterns beyond those previously mentioned, and I summarize them all here:
– excepting the special case of P1 = 2, the number of repeating digits for the n-th prime, Pn, is (Pn-1)
– the repeating groups of digits in each case have symmetry, in that the second half of the group is either identical to the first half, or, more commonly in these sets, is the inverse of the first half (e.g. 11101 in place of 00010)
– in most cases, 1/2 of all digits are 1. This includes, by inspection, all cases where the second half of the repeating pattern is the inverse of the first half, and some of the cases where the first and second halves are the same (e.g. 1/17 = 0.0000111100001111….
– the pattern always begins with “0” and ends with “1.” By inspection the first MUST be true. It’s not obvious why the second would necessarily always be true, and it may not, but it is for all cases examined so far
One may consider further that the only “important” digits after the binary point are those between the k-th and 2k-th positions after the binary point, where k is the position after the binary point of the first 1. This is true because the given prime, Pn, must be greater than 2^k, and not greater than [2^(k+1) – 2]. If the repeating pattern is truncated after the 2k-th binary digit and inverted and rounded, it will yield the prime number under consideration.
By “important” I only mean that one can obtain the prime number under consideration with only these digits. There may be other combinations of these same 2k digits that would lead to the same result, but if we know these first 2k digits (where the first k-1 are all 0, by definition, so we need only know k+1 digits), we can calculate the respective inverse prime number.
Here are the primes to 67 in illustration of this idea: