## Unique period primes

Proof of Analysis Primes

During my delving into old work to find my Ramanujan-type formula I came across a category of primes I had copied from Wikipedia in 2004, the ‘unique period primes’, totally forgotten about, but they are a further illustration of the fundamental logic of the repunits and their inner structure, they are the proof of my ‘Origin of the Primes’, in a sense.

3, 11, 37, 101, 9091, 9901, 333667, 909091,
99990001, 999999000001, 9999999900000001, 909090909090909091,
1111111111111111111, 11111111111111111111111, 900900900900990990990991,
909090909090909090909090909091

We immediately recognize the unique period primes as factors of the early repunits and these are very rare numbers, indeed. It are always deep-structure sequences of the repunits like 11, 111…111, 101, 100….001, 3, 37, 333667, 333……667, [90.90909… <91 (7×13), 9…1], 9091, 9090……9091, 99990001, 999900009999……….000099990001 etc. and sometimes even the structure of each others reciprocal.

In 2004 only 18 were known, now many more, but most have these specific structures with ones, zeros and nines, or must be repunits themselves, and come obviously straight from the repunit deep-structure.
Recently an ‘anomalous’ unique prime number was found which relates to the reciprocal of 7 and could indicate some elusive seven-number system hidden in the primes as it is in the irrational square root of 2.

When you see the list of unique primes, which I am unable to reproduce here in full, you will immediately recognize the regularities of the repunits and their deep-structure, which I gave in ‘Origin of the primes’.
Wikipedia gives an up to date list of the now proven ‘unique primes’, under that name.

It is not just a random number, this unique period prime, nearly all these ‘unique’ numbers have some obvious structure to themselves and this is in nearly all cases related to the deep-structure of the Repunit-Juggernaut. (This word popped up and is meant to express that the repunits ‘crush the freedom’ of the primes, because their supposed randomness turns out to be an ironcast structure, but the word Juggernaut derives originally from the Sanskrit Jagannath, which is the ‘Lord of the Universe’. Maybe it is blasphemous to state, but the repunits are the jewels in the crown of this Lord Jagannath, I think).

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