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User:Daniel Joyce/SequentialPrimes
Principle:
Use RSA-2048 from this web site as the catalyst for the probable prime divisors that are in the two OEIS wiki b-files also listed below.
http://en.wikipedia.org/wiki/RSA-2048#RSA-2048
These are the largest known sequential list of probable primes in descending order that are used as divisors of rsa2048. These two lists of primes are (almost factors) of rsa2048 that fall on the same triangle index of rsa2048 when the larger prime, in the list, is divided into RSA-2048 then multiplied with the resulting quotient creating a semi-prime > rea2048 but on the same triangle index as rsa-2048. The other list represents semi-primes < rsa-2048.
The first and second lists just shows the larger primes but there is also the quotient primes created by the division of the larger prime in both lists. These largest prime divisors were created from the smaller prime divisors as next prime function in Python that's why the sequence of largest primes are in descending order in both lists.
The last prime listed in this first list along with it's smaller prime counter-part create a semi-prime record that is the closest to rsa2048 and also > rsa2048 but with a record smallest gap between rsa-2048 and this semi-prime of 1.1911..e+159
Meaning the most 9's in the remainder when either prime is divided into rsa2048. By truncating the remainder and adding (1) creates the quotient prime. As this sequence continues beyond the record gap above, the semi-primes created from these two primes become < rsa2048. and thus switching to remaining zeros in the quotient prime.(Second b-file)
These primes are polynomial in relation to rsa2048 where the smaller quotient prime can add +8+8+8+.. or -8-8-8 and at any point of these summations or negations the integer at that location could be prime but most likely composite.
The larger prime is +9+9+9+.. or -9-9-9-9.. but in order to keep the integers odd it is +18+18+18+.. or -18-18-18-18. So cherry picking from both large and small primes you will finally get a match where both are prime. For each smaller prime, between 5 and 10 comparisons of next prime function (larger prime) has to be made. So the algorithm is slow before you get a prime matching hit.
If you wish to make this list longer and continue the search just for the two prime pairs do a multiple add of 18 to the first prime in the first list. Like 18*10^20 + first prime in list. This will add to the beginning of the first prime list but will never reach the beginning of the first list because of trillions and trillions of matched primes that will occur with an almost uncountable number of calculations. Divide this integer from above into rsa2048 and add one to the quotient and truncate the 9's in the smaller divisor. Use this smallest divisor in the next prime function and then use that prime to find the next prime for the larger prime. If no match find the next smaller prime and so on. Continuing the prime search with the smaller prime and when you get a prime hit with the matching larger prime you have a pair of primes creating a semi-prime still way into the range of being on the same triangle index of rsa2048. To just get close to the index triangle number and still remain on the index you would have to add to the first prime in the list something like 18*10^151 and it would still fall on the same index as rsa2048. So the potential for many trillions of prime match ups in this calculation is not out of the question when used in relation to rsa2048.
I have created many other discrete super-cycle list like this one starting with larger and smaller primes and like here, using rsa2048. You can use any large composite other then rea2048 or even a large 2048 bit prime.
Calculating these super-cycles is accomplished by, in this case the ratio(1.125), sqrt(rsa-2048)/sqrt(1.125) and then reaching the first zero in the remainder of the quotient by adding or subtracting e+154 or e+153 to the smallest divisor. When the first zero remainder is attained in the larger quotient then add, if that was the case to get the first zero, e+153 or e+152 otherwise subtract e+153 or e+152 always dropping the amount as each remainder zero is added to the larger quotient. A tedious process but attainable. There are some complex shortcuts but will leave it at that.
I kept the first list short and also the second list to be reasonable but have created thousands of semi-primes from other super-cycle ratios.
Link to the list (First b-file).
The next list is the start of zero remainders in the quotient but still a part of this super cycle. Making the first prime in this second list the transition from nine's ,last list, quotient remainder too the start of zero quotient remainders in the second list below.
Link to the List(Second b-file) [(User:Daniel_Joyce/StartZerosdfile.txt|list)] (Second b-file).--Daniel Joyce 15:58, 19 August 2013 (UTC)