Recursive prime p(k+1)=m*p(k)^2+/-1 with minimum m

By: | Post date: August 30, 2010 | Comments: 6 Comments

Posted in categories: Fun Stuffs, Prime Search

Define p(0)=1;
p(1)=2*P(0)^2+1=3;
p(2)=2*p(1)^2-1=17;
p(3)=2*p(2)^2-1=577;
p(4)=2*p(3)^2-1=665857;
p(5)=20*p(4)^2-1;
p(6)=2*p(5)^2-1;
p(7)=28*p(6)^2+1;
p(8)=182*p(7)^2-1;
p(9)=272*p(8)^2-1;
p(10)=540*p(9)^2+1;
p(11)=162*p(10)^2+1;
p(12)=1002*p(11)^2+1;
p(13)=112*p(12)^2+1;
p(14)=306*p(13)^2-1;
p(15)=1752*p(14)^2+1;
p(16)=20564*p(15)^2-1;
p(17)=135236*p(16)^2-1;
p(18)=547952*p(17)^2-1;
p(19)=282904*p(18)^2+1;

p(19)=282904*(547952*(135236*(20564*(1752*(306*(112*(1002*(162*(540*(272*(182*(28*(2*8867310888979^2-1)^2+1)^2-1)^2-1)^2+1)^2+1)^2+1)^2+1)^2-1)^2+1)^2-1)^2-1)^2-1)^2+1 has database ID 94235 in The List of Largest Known Primes Home Page. The direct link is HERE.

These primes are recursively proven using OpenPFGW, by the command
pfgw -t (or tp) -h”p(k)” p(k+1)
The number
p(k+1)=m*p(k)^2+/-1
are reformatted by Mathematica to get the short expression.

The final certification code is
#!/bin/sh
./pfgw -l”pmrtrain.21.cert” -t p_06
./pfgw -l”pmrtrain.21.cert” -tp -h”p_06″ p_07
./pfgw -l”pmrtrain.21.cert” -tp -h”p_07″ p_08
./pfgw -l”pmrtrain.21.cert” -tp -h”p_08″ p_09
./pfgw -l”pmrtrain.21.cert” -tp -h”p_09″ p_10
./pfgw -l”pmrtrain.21.cert” -t -h”p_10″ p_11
./pfgw -l”pmrtrain.21.cert” -t -h”p_11″ p_12
./pfgw -l”pmrtrain.21.cert” -t -h”p_12″ p_13
./pfgw -l”pmrtrain.21.cert” -t -h”p_13″ p_14
./pfgw -l”pmrtrain.21.cert” -t -h”p_14″ p_15
./pfgw -l”pmrtrain.21.cert” -t -h”p_15″ p_16
./pfgw -l”pmrtrain.21.cert” -t -h”p_16″ p_17
./pfgw -l”pmrtrain.21.cert” -tp -h”p_17″ p_18
./pfgw -l”pmrtrain.21.cert” -tp -h”p_18″ p_19
./pfgw -l”pmrtrain.21.cert” -t -h”p_19″ p_20

The certificate will be posted when done.

6 Comments

  • Manda Sockalosky says:
    January 21, 2011 at 10:19 pm

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    November 27, 2010 at 6:49 am

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  • magicpeople says:
    September 12, 2010 at 10:00 am

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  • makaronnik says:
    September 12, 2010 at 8:52 am

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  • leizhou says:
    August 31, 2010 at 8:20 am

    Proof:
    Primality testing (665857)^2*20-1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 2
    Calling Brillhart-Lehmer-Selfridge with factored part 55.81%
    (665857)^2*20-1 is prime! (0.0013s+0.0004s)
    Primality testing ((665857)^2*20-1)^2*2-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 7, base 6+sqrt(7)
    Calling Brillhart-Lehmer-Selfridge with factored part 98.85%
    ((665857)^2*20-1)^2*2-1 is prime! (0.0032s+0.0003s)
    Primality testing (((665857)^2*20-1)^2*2-1)^2*28+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 2
    Calling Brillhart-Lehmer-Selfridge with factored part 97.75%
    (((665857)^2*20-1)^2*2-1)^2*28+1 is prime! (0.0019s+0.0002s)
    Primality testing ((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 7, base 1+sqrt(7)
    Calling Brillhart-Lehmer-Selfridge with factored part 97.81%
    ((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1 is prime! (0.0069s+0.0003s)
    Primality testing (((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 3, base 1+sqrt(3)
    Calling Brillhart-Lehmer-Selfridge with factored part 98.92%
    (((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1 is prime! (0.0126s+0.0002s)
    Primality testing ((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 2
    Calling Brillhart-Lehmer-Selfridge with factored part 99.39%
    ((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1 is prime! (0.0157s+0.0003s)
    Primality testing (((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 2
    Calling Brillhart-Lehmer-Selfridge with factored part 99.77%
    (((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1 is prime! (0.0530s+0.0002s)
    Primality testing ((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 2
    Calling Brillhart-Lehmer-Selfridge with factored part 99.83%
    ((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1 is prime! (0.2114s+0.0003s)
    Primality testing (((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 3
    Calling Brillhart-Lehmer-Selfridge with factored part 99.94%
    (((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1 is prime! (0.8421s+0.0004s)
    Primality testing ((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 3, base 3+sqrt(3)
    Calling Brillhart-Lehmer-Selfridge with factored part 99.97%
    ((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1 is prime! (12.3430s+0.0006s)
    Primality testing (((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 7
    Calling Brillhart-Lehmer-Selfridge with factored part 99.98%
    (((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1 is prime! (16.8592s+0.0011s)
    Primality testing ((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 3, base 4+sqrt(3)
    Calling Brillhart-Lehmer-Selfridge with factored part 99.98%
    ((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1 is prime! (235.2879s+0.0019s)
    Primality testing (((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 3, base 4+sqrt(3)
    Calling Brillhart-Lehmer-Selfridge with factored part 99.99%
    (((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1 is prime! (956.3106s+0.0056s)
    Primality testing ((((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1)^2*547952-1 [N+1, Brillhart-Lehmer-Selfridge]
    Running N+1 test using discriminant 3, base 1+sqrt(3)
    Calling Brillhart-Lehmer-Selfridge with factored part 100.00%
    ((((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1)^2*547952-1 is prime! (4246.8061s+0.0101s)
    Primality testing (((((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1)^2*547952-1)^2*282904+1 [N-1, Brillhart-Lehmer-Selfridge]
    Running N-1 test using base 3
    Calling Brillhart-Lehmer-Selfridge with factored part 100.00%
    (((((((((((((((665857)^2*20-1)^2*2-1)^2*28+1)^2*182-1)^2*272-1)^2*540+1)^2*162+1)^2*1002+1)^2*112+1)^2*306-1)^2*1752+1)^2*20564-1)^2*135236-1)^2*547952-1)^2*282904+1 is prime! (5628.6941s+0.0236s)

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