## Repunits and primes

Orkney 2010

This text below is the integral text which I sent 6 years ago to Nature, CWI Holland, and QI Ltd, Oxford. I have only omitted one sentence which is obviously wrong, but not essential, so as to keep the flow of the argument unhindered by an obviously wrong side remark. The sentence is kept in the other version, the original, which has the text precisely as it was when sent off.

I am making a different text at the moment, but this text below has all essential arguments in it, I only found new evidence of another layer of order, which shows again that no prime can go it alone, they are all absolutely locked in via their reciprocals, which is a more general concept than the repunits, although it is about exactly the same thing.

I will probably publish more text from years ago about this subject because it is much more than just finding the order of the primes, this is only the beginning.

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Orkney 2010 / Oostvoorne,  2/4/2004

Repunits , the order of the primes

It is held here that the order of the primes is a logical necessity, that no calculation is needed for its proof, only reasoning and the argument is as follows :

All primes have a period in the inverse in the decimal system, except 2 and 5 (exc 2&5) which produce the decimal system

This period will never end in a zero, an even number or a 5, like the primes themselves

The period signifies that there is an endless repeated division, that results in the same number series over and over again

This series of digits has a certain length that characterizes the prime, but the number of digits in this period length can never exceed the number of the prime minus 1, and that is a mathematical necessity for any period

The period repeats when the division has reached a remainder of 1 and the division starts all over again, ad infinitum, this is the periodic characteristic of the prime’s reciprocal (exc 2&5)

This entails that the product of prime and period are always (10^n) – 1 or 99……99 , a nine-only number

Since the nine-only number, – the product of prime and period – , is always divisible by 9 and then results in a repunit (repeated unit) , a one-only number, 11…….11 , symbol R(n) and since the prime is never divisible (by 9) by definition , it must be the period that is always divisible by 9

This entails that the period divided by 9 results in a number that is the ‘deep-structure period’ of the prime, because this number, hidden behind the (trivial) factor 9, is either a prime or a product of primes (exc 2&5 ) and forms, multiplied by the prime, the repunit

Since the period can never exceed the number of the prime, we know the repunit the prime belongs to has at most 1 digit less than the number of the prime, R (p – 1)

The initial prime, by its inverse, is linked to a fixed set of other primes, that are all factors to each others periods and, multiplied, together form this very peculiar product, the repunit

This is the deep structure of the primes, the ‘repunit of first occurrence’, (rofo) , the ring of numbers that always interlock and keep together in a set frame

So every prime is either a factor to a repunit or a repunit itself

Repunits that are prime are relatively rare, but like R2, R19, R23, R317 , R1031, suggest another layer may be behind it (because 317 = 3…7 and 1031 = 10…1, and also 17 and 31 seem involved)

So all repunits are either prime or a unique product of primes, every next repunit revealing at least one new prime (when prime itself), but mostly two or (many) more other ‘new’ primes

New primes in the factorisation of the repunit are always multiples of its ordinal number + 1

All repunits hold all the primes as factors (exc 2&5)

This is the logical necessity of primes and repunits

The repunits hold the order of the primes QED

P.S.
the computational proof of this order is found by taking any prime
form its largest possible ‘repunit of first occurrence’, which is R (prime – 1)
all primes less than the chosen prime will be among the factors to all repunits less than the calculated repunit R(prime – 1)
this net catches all