Lucas-Lehmer Primes

[Batalov]

And one for n=16, with s₀=17073:

(((((((((((((((17073^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2
or, slightly shorter written, =
((((((((((((((291487327^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2)^2-2
(277369 digit, PRP)

[paulunderwood]

Congrats. Have you done a Lucas PRP test on these (3) Fermat PRPs?

[Batalov]

Sure did.

[science_man_88]

Originally Posted by Xyzzy

It took us a while but we think we have this right:

30201 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 - 2 ^ 2 -

We really dislike parentheses!

sorry for awakening this thread again ( there's a new similar thread)
well they are mathematically meaningful as they allow proper values without parentheses yours is equivalent to 30201 -4-4-4-4-4-4-4-4-4-4-4-4-4-4 if it was completely done properly in theory though I think you misplaced the operators by having a space in between things as well.

[henryzz]

My testing on n=17 has now passed x=1000. Tests now take around an hour.

[Batalov]

Would you like to compare residues ?
(I didn't plan to run n=17 but then left some running on a laptop, so I have some residues up to 3000, I think...)

[henryzz]

Originally Posted by Batalov

Would you like to compare residues ?
(I didn't plan to run n=17 but then left some running on a laptop, so I have some residues up to 3000, I think...)

I am fairly confident in the pc I am running these on. Testing should speed up shortly as I will be using a sieved file produced on my slower Linux machine(that spends a lot of time in windows). Previously I have just left it running using pfgw factoring.
I would gladly compare them though if you have them available.

[henryzz]

I tested to n=17 to 4k. Stopping for now.

[Batalov]

Here is something that might come as a surprise.

1. What if you take L=3, and then repeat Lucas transformation L=L^2-2... do you know what you will get? You will get normal Lucas numbers L(2^n).

2. Can these be prime? Yes! Have a look at OEIS oeis.org/A001606; primes and powers of two are eligible!

3. What does this code do?

$ gp
for(n=10,66,m=2*2^n;forstep(f=m-1,2^(n+20),[m+2,m-2],if(!isprime(f),next);L=Mod(3,f);for(k=2,n,L=L^2-2);if(!L,print(f" | L(2^"n")"))))
65241089 | L(2^12)
524287 | L(2^18)
16074670081 | L(2^19)
1811939329 | L(2^22)
167772161 | L(2^23)
1176482497601470463 | L(2^24)
3758096383 | L(2^28)
15868293545983 | L(2^28)
2147483647 | L(2^30)
206158430209 | L(2^31)
5703716569087 | L(2^35)
17575814316278415361 | L(2^39)
7595426324676607 | L(2^41)
79543069199826943 | L(2^42)
609885905787813887 | L(2^45)
3204381503618285567 | L(2^45)
55164591835661205503 | L(2^51)
9400813862173173350401 | L(2^53)
166453042227613532161 | L(2^54)
264028247927004812279807 | L(2^63)

4. While we can't practically approach L(2^29) character, we can play with L(2^24), L(2^25), L(2^26), L(2^27). All lesser are known to be composite, except L(2^1), L(2^2), L(2^3), L(2^4). Just like Fermat primes! :-)

P.S. 1176482497601470463 | L(2^24) using LLPsieve, so L(2^24) is off the list.

[sweety439]

The index of the first prime in some Lucas sequences or Lehmer sequences are listed in these sequences: A269254, A269252 and A305534.