Leyland Primes: ECPP proofs

[sweety439]

Originally Posted by storm5510

The above only took a few seconds. You must be running it in a different way?

A number pass a primality test (e.g. Fermat primality test, strong primality test, Lucas strong primality test, strong Lucas primality test, …) need not be prime, for the smallest composites that pass the strong primality tests to the first n prime bases see https://oeis.org/A014233

To completely solve a problem, we need to 100% sure that the number is actually prime, and to be 100% sure of primality we need to follow a method that actually proves the number prime. However, currently for large numbers, only the numbers with N-1 and/or N+1 can be >= 1/3 factored can be proven to be primes, by using N-1 primality proving or N+1 primality proving, for other numbers, we usually use ECPP primality proving such as PRIMO, e.g. for the minimal strings for the primes > b in base b problem, the cases that b = 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 24 are completely solved (they have 1, 3, 5, 22, 11, 71, 75, 151, 77, 106, 650, 1284, 549, 3314, 3409 minimal strings for the primes > b, respectively, and the width of the longest minimal string for the primes > b for these bases b are 2, 3, 3, 96, 5, 17, 221, 1161, 31, 42, 19699, 157, 6271, 6271, 8134, respectively), since for these bases b, all minimal strings for the primes > b in base b are found and it is proved that there cannot be any other minimal strings for the primes > b in base b (since it is proved that all numbers > b not containing any of these strings as subsequence in base b, are composite), also all of these primes are proven primes (i.e. we are 100% sure that these numbers are actually primes), and not merely probable primes, e.g. this is the primality certificate for the largest minimal string for the primes > b for base b = 24, although for this prime neither N-1 nor N+1 can be easily >= 1/3 factored, but for the cases that b = 11, 22, 30, all minimal strings for the primes > b in base b are found and it is proved that there cannot be any other minimal strings for the primes > b in base b (since it is proved that all numbers > b not containing any of these strings as subsequence in base b, are composite), but some of these primes are only probable primes (they are: 5(7^62668) in base 11, B(K^22001)5 in base 22, I(0^24608)D in base 30), thus we cannot definitely say that these three bases are solved, and we cannot definitely say base 11 has 1068 minimal strings for the primes (it is 1067 if 5(7^62668) in base 11 is in fact composite and no prime of the form 5777…777 in base 11 exists) and the width of the longest minimal string for the primes > b for base b = 11 is 62669 (it is > 62669 if 5(7^62668) in base 11 is in fact composite but a prime of the form 5777…777 in base 11 exists, and it is 1013 if 5(7^62668) in base 11 is in fact composite and no prime of the form 5777…777 in base 11 exists), although 5(7^62668) in base 11, B(K^22001)5 in base 22, I(0^24608)D in base 30, are strong PRP to all prime bases <= 61 and strong Lucas PRP with parameters (P, Q) defined by Selfridge's Method A and trial factored to 10^11 (I want to trial factor them to 10^16, but PFGW cannot handle that large trial factor limit, so how to do this?)

[paulunderwood]

Originally Posted by sweety439

However, currently for large numbers, only the numbers with N-1 and/or N+1 can be >= 1/3 factored can be proven to be primes, by using N-1 primality proving or N+1 primality proving, for other numbers, we usually use ECPP primality proving such as PRIMO, e.g. for the minimal strings for the primes > b in base b problem,

sweety439 fails to mention the 30% of KP and 25%+epsilon of CGH. Also, Primo is currently old hat and people are using FastECPP/CM.

[sweety439]

Originally Posted by paulunderwood

sweety439 fails to mention the 30% of KP and 25%+epsilon of CGH. Also, Primo is currently old hat and people are using FastECPP/CM.

I hope that factordb can add them.