20150827, 18:20  #1 
Feb 2004
France
39A_{16} Posts 
OEIS  A059242 : 2^n+5 , n odd
See: https://oeis.org/A059242 .
The LLTlike algorithm, base on a cycle, is described by this PARI/gp program: Code:
for(i=1,100000, n=2*i+1 ; N=2^n+5 ; x=194 ; for(j=1, n2, x=Mod(x^22,N)) ; if(x==194, print(n," Prime") ) ) Please help checking that the algorithm still produces good results with higher values of n. No false positive, no false negative. Up to now, it is perfect ! So, here is a new LLTlike algorithm that could lead, when using fast implementation of LLT, to new BIG PRPs ! Last fiddled with by T.Rex on 20150827 at 18:23 
20150827, 18:22  #2  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Code:
a=[];forstep(i=100000,1,1,n=i<<1+1;N=1<<n+5;x=Mod(194,N);forstep(j=n,3,1,x=sqr(x)2);if(x==194,a=concat(a,n)));print(a",prime") Last fiddled with by science_man_88 on 20150827 at 18:22 

20150827, 18:46  #3 
Feb 2004
France
2×461 Posts 

20150827, 18:52  #4  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:
Code:
(13:12) gp > a=[];forstep(i=1183,1,1,n=i<<1+1;N=1<<n+3;x=Mod(14,N);forstep(j=n,2,1,x=sqr(x)2);if(x==14,a=concat(a,n)));print(a",prime") [2367, 2191, 2139, 67, 55, 15, 7, 3],prime (14:36) gp > ## *** last result computed in 3,088 ms. (14:36) gp > for(i=1,1183,n=2*i+1;N=2^n+3;x=14;for(j=1,n1,x=Mod(x^22,N));if(x==14,print(n," Prime"))) 3 Prime 7 Prime 15 Prime 55 Prime 67 Prime 2139 Prime 2191 Prime 2367 Prime (14:36) gp > ## *** last result computed in 3,220 ms. is a comparison form your older codes in the other thread(s) though I did a loop outside them to check using 100 times completing the task to my mark and it only resulted in a small savings of 20 seconds ( the 100 rounds of it aka about 200 ms per time it ran up to these limits) I think. edit2: admittedly this is on a machine with windows 10 and version 2.7.4 64 bit version. Last fiddled with by science_man_88 on 20150827 at 18:59 

20150828, 07:11  #5 
Feb 2004
France
2×461 Posts 
16541
OK. My PARI/gp code found 16541, which also is in A059242.
It is not useful to let PARI/gp continue searching, since a much faster implementation of LLT must be used in order to find a n greater than 282203 such that N=2^n+5 is a biggest PRP . I remind people that LLTlike algorithms are the fastest math ways for proving that a number N is Prime or PRP., when such an algorithm has been found by experiment (PRP, like for Wagstaff numbers) or proven (Prime, like for Mersenne or Fermat numbers). But one has to adapt existing LLTcode to the specific number N. 
20150828, 12:25  #6  
Einyen
Dec 2003
Denmark
3^{2}·5·71 Posts 
The test is positive for 11, 47, 53, 141, 143, 191, 273, 341, 16541, 34001, 34763, 42167, 193965, 282203, and no false positives up to 22.5k
Quote:
2^3+5 = 13, S0=194, S1=12. 2^5+5 = 37. S0=194, S1=5, S2=23, S3=9. Last fiddled with by ATH on 20150828 at 12:25 

20150828, 13:18  #7 
Jun 2003
3·11·157 Posts 

20150828, 18:04  #8 
Einyen
Dec 2003
Denmark
3^{2}×5×71 Posts 

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