# Archive:

## Month: December 2010

## 19th Recursive prime in the form of p[k+1]=l(k+1)*(m[k]^2-n[k]^2)+/-1, while p[k]=m[k]+n[k]

Define: p[1] = m[1] + n[1] = 2 + 1 p[2](l=2) = 2*(m[1]^2-n[1]^2)-1 = 2*(4-1)-1 = 5 = 2^3 – 3 p[3](l=2) = 2*(2^6-3^2)-1 = 2^7 – 19 p[4](l=4) = 4*(2^14 – 19^2) -1 = 2^16 – 1445 p[5](l=8) = 8*(2^32 – 1445^2) +1 = 2^35 – 16704199 p[6](l=18) = 2^71*3^2 – (16704199^2*18+1) p[7](l=84) = […]

## New prime found: 2^74015*3^42*35^8*5863^16*137^73952*953^2*1223^4*15217 – ((((3643*121875747021497257)^2*19568-1)^2*7624+1)^2*273906-1)

Using Primo, 274^2311 – 83 is proven a prime number. Define this as p[1]=a[1]-b[1], while a[1]=274^2311 and b[1]=83. p[2]=16236*(a[1]^2-b[1]^2)-1 =16236*274^4622 – 111849805 is proven prime using pfgw: pfgw -h”p[1]” -tp “p[2]” Keep going in this way, it is obtained: p[3]=2^9249*3^5*5*7*11^2*13^2*41^2*137^9244 – (3643*121875747021497257) p[4]=2^18502*3^10*5^2*7^2*11^4*13^4*41^4*137^18488*1223 – ((3643*121875747021497257)^2*19568-1) p[5]=2^37007*3^20*35^4*5863^8*137^36976*953*1223^2 – (((3643*121875747021497257)^2*19568-1)^2*7624+1) p[6]=2^74015*3^42*35^8*5863^16*137^73952*953^2*1223^4*15217 – ((((3643*121875747021497257)^2*19568-1)^2*7624+1)^2*273906-1) Certificate will be posted […]

## Recursive prime brother by Brillhart – Lehmer – Selfridge algorithm

Define: p[k,i]=ABS[1+2*n[k,i]*p[k-1,1]*p[k-1,2]],n[k,1] is the integer with minimum ABS[n[k,1]] that makes p[k,1] a prime number, and n[k,2] is the integer with second minimum ABS[n[k,2]] that makes p[k,2] a prime number The primality of p[k,i] can be proven using Brillhart – Lehmer – Selfridge algorithm recursively by using p[k-1,1] and p[k-1,2] as helper since n is a […]