Primality Criterion for F_n(132)

1. Feb 22, 2012

pedja

Primality Criteria for F_n(132)

$$\text{Let's define sequence}~ S_i ~\text{as :}$$
$$S_i= T_{66}(S_{i-1})=2^{-1}\cdot \left(\left(S_{i-1}+\sqrt{S_{i-1}^2-1}\right)^{66}+\left(S_{i-1}-\sqrt{S_{i-1}^2-1}\right)^{66}\right) , ~\text{with}~ S_0=8$$
$$\text{and define} ~F_n(132)=132^{2^n}+1$$

$$\text{I found that :} ~F_2(132) \mid S_5 , ~ F_3(132) \mid S_{13} , ~F_5(132) \mid S_{61}$$

How to prove following statement :

Conjecture :
$$F_n(132) ;~ (n\geq 1)~\text{ is a prime iff}~F_n(132) \mid S_{2^{n+1}-3}$$

Last edited: Feb 22, 2012
2. Feb 23, 2012

Norwegian

Hi Pedja. I am somewhat curious about this. Interesting conjecture, but the numbers Si are a bit too big for my taste. Have I got it right when I say that your S61 is slightly larger than a googolplex?

What makes you particularly interested in the numbers 66 and 132. Do you have any reasons to believe your assertion, other than the relations you mention?

3. Feb 23, 2012

pedja

Hi . You don't have to calculate value of $$S_{2^{n+1}-3}$$ to find out whether $$F_n(132) \mid S_{2^{n+1}-3}$$ See Wikipedia article : Lucas-Lehmer primality test

One can formulate similar conjectures for other Generalized Fermat numbers .

Primality test based on this conjecture written in Mathematica :
Code (Text):

n = 3;
GF = 132^(2^n) + 1;
For[i = 1; s = 8, i <= 2^(n + 1) - 3, i++,
s = Mod[-1 + 114270464*s^6 + 420384712704*s^10 -
13554222252032*s^12 + 313683429261312*s^14 -
5437179440529408*s^16 + 72851097078988800*s^18 -
772988482690744320*s^20 + 6618923024944988160*s^22 -
46428387595266293760*s^24 + 269998930938625523712*s^26 -
1314280510389076623360*s^28 + 5396103428861818044416*s^30 +
2178*s^2 - 789888*s^4 - 8815150080*s^8 -
18799328074744398348288*s^32 + 55828307615907607216128*s^34 -
141786178072146304040960*s^36 + 308581582432978442649600*s^38 -
576018953874893092945920*s^40 + 921897930824161070940160*s^42 -
1262980674786588528476160*s^44 +
1476528131876108327976960*s^46 -
1466056301153582736998400*s^48 +
1227896951007000725028864*s^50 - 859342662544869285691392*s^52 +
496028678729603095724032*s^54 - 231909512133320927870976*s^56 +
85580642711025871749120*s^58 - 23981920217327023947776*s^60 +
4793847616155269726208*s^62 - 608742554432415203328*s^64 +
36893488147419103232*s^66, GF]];
If[s == 0, Print["prime"], Print["composite"]];

Last edited: Feb 23, 2012