Recursive prime triplet by Brillhart – Lehmer – Selfridge algorithm
Take any three primes, say p[1,0], p[2,0], and p[3,0].
Define:
p[i,j]=ABS[1+2*n[i,j]*p[(i+1) mod 3,j-1]*p[(i+2) mod 3,j-1]],n is the integer with minimum ABS[n] that makes p[i,j] a prime number.
The primality of p[i,j] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[(i+1) mod 3,j-1] and p[(i+2) mod 3,j-1] as helper since n is a small integer, by reducing j to 0.
With this idea, taking
p[1,0]=3, p[2,0]=5, p[3,0]=7
We got the n[i,j] ( columns : j; rows: i):
| i | j=1 | j=2 | j=3 |
|---|---|---|---|
| 1 | 1 | -1 | -1 |
| 2 | -1 | 2 | -1 |
| 3 | -8 | -10 | 7 |
| 4 | -14 | -3 | -13 |
| 5 | -18 | 24 | 46 |
| 6 | 24 | 39 | -32 |
| 7 | 225 | -48 | 27 |
| 8 | 120 | -76 | 30 |
| 9 | -132 | 245 | -676 |
| 10 | 316 | -722 | 65 |
| 11 | 55 | -1197 | -510 |
| 12 | -427 | -1716 | -637 |
| 13 | 4651 | -1158 | 3420 |
| 14 | -16337 | 17640 | -18426 |
| 15 | -8915 | -70649 | -31489 |
| 16 | -18844 | -92841 | 124053 |
| 17 | -144011 | -8853 | -14042 |
The last three, p[i,17] makes top 5000 list.
The proof will be posted in the reply of this one.
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Slam dunkin like Shaquille O’Neal, if he wrote informaivte articles.
It’s much easier to undertsand when you put it that way!
These four primes are believed to be rank 18 primes. For the definition of prime ranking, please see http://bitc.bme.emory.edu/~lzhou/blogs/?p=117
For these primes, no existing primality proving program can prove them in a single step. OpenPFGW can only do PRP on them. And it is far beyond the capability of ECPP method. To prove them, the only way is to do it recursively. First, prove the primality of small j (j <=4) using pfgw. Then use p[i,j] (i=1,2,3) as helpers to prove the primality of p[i,j+1] (i=1,2,3), until the latest j you have reach.
Also, these primes does not have a known way to be written into an expression that is shorter than its full digits format.
Currently for seed 3,5,7, I have reached j=17 and found 4 blob primes that make into the top 500 prime list.
They are:
p[1,17]=288022*p[2,16]*p[3,16]-1
p[2,17]=17706*p[3,16]*p[1,16]-1
p[3,17]=28084*p[1,16]*p[2,16]-1
20356*p[3,16]*p[1,16]+1
All helper files are packed into this file.
Also 20356*p[3,16]*p[1,16]+1 is proven a prime while searching for p[2,17].
The proof can be found at HERE
Full certificate please see pmtrio_3_5_7.18.cert