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-   -   Repeating residues in LL tests of composite Mersenne numbers (https://www.mersenneforum.org/showthread.php?t=25772)

Viliam Furik 2020-07-24 20:10

Repeating residues in LL tests of composite Mersenne numbers
 
I have noticed, that when doing LL tests far behind the p-2 iteration, residues start to repeat with a certain period. This happens only for composite Mersenne numbers because when prime ones hit 0, next iteration yields 2^p-2, and then the residue is 2 all the way up to infinity.

I found periods for four composite Mersenne numbers, using my rather slow Python script:

M11 - 60 iters
M23 - 32340 iters
M29 - 252 iters
M37 - ??? iters (did not finish in my patience time)
M41 - 822960 iters

I wonder what's the reason behind these periods. Is there some formula for them? Can this be used somehow for testing or factoring purposes? Is it even known fact?

My best guess is that it's the number of quadratic residues modulo Mersenne number.

Viliam Furik 2020-07-25 10:00

[QUOTE=Viliam Furik;551475]...when prime ones hit 0, next iteration yields 2^p-2, and then the residue is 2...
[/QUOTE]
It should be 2^p - 3, or Mp - 2.

preda 2020-08-01 07:52

[QUOTE=Viliam Furik;551475]I have noticed, that when doing LL tests far behind the p-2 iteration, residues start to repeat with a certain period. This happens only for composite Mersenne numbers because when prime ones hit 0, next iteration yields 2^p-2, and then the residue is 2 all the way up to infinity.

I found periods for four composite Mersenne numbers, using my rather slow Python script:

M11 - 60 iters
M23 - 32340 iters
M29 - 252 iters
M37 - ??? iters (did not finish in my patience time)
M41 - 822960 iters

I wonder what's the reason behind these periods. Is there some formula for them? Can this be used somehow for testing or factoring purposes? Is it even known fact?

My best guess is that it's the number of quadratic residues modulo Mersenne number.[/QUOTE]

Hi Viliam, I'm by no means an LL expert, and I don't have an answer to your question here. However, I would approach the question by drilling down into one (or a few) mathematical proofs of the LL test, to understand why it works and the underlying mechanics. Might be that what we see is the "multiplicative order" in some group on which LL is based.. anyway somebody more maths-grounded could probably easily shed light here.

JeppeSN 2020-08-02 09:26

The LL sequence starts 4, 14, 194, ...

Since we calculate modulo 2^p - 1, there are only finitely many values we can hit, so sooner or later we are going to hit a value we have seen before.

For 2^p - 1 prime, we know we hit 0, "-2", 2, 2, 2, ... So the period starts late and has length 1.

For 2^p - 1 composite (p prime), do you know if it is always 14 which is the first term to reappear? How does your script detect that a period has finished (in order to report the period length)?

/JeppeSN

Viliam Furik 2020-08-02 12:50

[QUOTE=JeppeSN;552287]
For 2^p - 1 composite (p prime), do you know if it is always 14 which is the first term to reappear? How does your script detect that a period has finished (in order to report the period length)?

/JeppeSN[/QUOTE]

It simply looks for a value 14, based on previous observation, that periodicity starts at first modular squaring (S(1) = 14). But to answer previou question, I don't actually know that for sure, it's just that I haven't found a counterexample yet.

Viliam Furik 2020-08-02 18:27

Python code
 
My slow but simple Python code in its entirity:
[CODE]s = 4
for a in range(2 ** 37 + 1):
s = (s ** 2 - 2) % (2 ** 37 - 1):
if s == 14:
print(a)[/CODE]

Batalov 2020-08-02 23:47

[CODE] period 60 for M11
period 32340 for M23
period 252 for M29
period 516924 for M37
period 822960 for M41
period 420 for M43
period 20338900 for M47
period 1309620 for M53
period 345603421440 for M59

(period is always a multiple of (p-1) )
_____________

main(int argc,char **argv) {
uint64_t f,b,c,j,k,l,l2;
//...

if(argc<=1) { printf(" Use: %s p\n\t (copyleft) 2020 S.Batalov\n",argv[0]); exit(0); }
if(argc>1) l = atoll(argv[1]);
printf("# %lld ... testing\n",l);
l = 1ULL << l;
l--;
c = 4;
for(f=1; f<=120000;f++) { # or some other number, -- to get into "real" cycle
c = mulmod(c,c,l)-2;
}
b = c;
c = mulmod(b,b,l)-2;
for(f=1; c!=b && f<=l;f++) {
c = mulmod(c,c,l)-2;
}
printf(" period %lld for M%lld\n",f,atoll(argv[1]));
exit(0);
}
[/CODE]
The cycle does not include 4 (of course*), and not 14 or even 194 for larger p values, but we can greedily simply scroll forward "far enough".

*we all already know that we will never see 4 again, because 6 is a nonresidue. 16 and 196 are residues

JeppeSN 2020-08-03 07:33

Thanks, Batalov, that confirms my suspicion. For example for p=37 (the first one Viliam Furik's method failed for), we start with:

4 -> 14 -> 194 -> 37634 -> 1416317954 -> (period starts here) 111419319480 -> ...

So in this case, the period starts at the first term after we have "wrapped around" because we work modulo 2^37 - 1 = 137438953471.

/JeppeSN

Viliam Furik 2020-08-03 07:54

[QUOTE=Batalov;552363][CODE] period 60 for M11
period 32340 for M23
period 252 for M29
period 516924 for M37
period 822960 for M41
period 420 for M43
period 20338900 for M47
period 1309620 for M53
period 345603421440 for M59

(period is always a multiple of (p-1) )
[/CODE]
[/QUOTE]
Спасибо! But it still leaves the question "Why that period?" hanging in the air...

Batalov 2020-08-03 08:35

We had a forum member about a decade ago who drilled into residue pathways/topology/cycles but he has left many years ago and left all that quite unfinished. (Tony "T.Rex")
He had some conjectures but as far as I remember those ended up broken.
(Including one about a Wagstaff number test.)

One can search forum way back, using Search / Advanced / Search by User Name, and search for his threads, such as DiGraph for LLT, LLT Cycles, LLT Tree... there is probably something of interest (just don't assume those things right ... or wrong), pick and chose what is useful.
[url]https://mersenneforum.org/showthread.php?t=5935[/url]
[url]https://mersenneforum.org/showthread.php?t=10670[/url]
...and more

Batalov 2020-08-03 16:45

[QUOTE=Viliam Furik;552390]Спасибо! But it still leaves the question "Why that period?" hanging in the air...[/QUOTE]

Just a wild guess that:
1. residue is in effect a Chinese Remainder of residues mod all factors of these (composite) Mp
2. this is an lcm() of individual periods guided by each factor.

[CODE]M11 = 23 · 89
M23 = 47 · 178481
M29 = 233 · 1103 · 2089
M37 = 223 · 616318177
M41 = 13367 · 164511353
M43 = 431 · 9719 · 2099863
M47 = 2351 · 4513 · 13264529
M53 = 6361 · 69431 · 20394401
M59 = 179951 · 3203431780337[/CODE]
Factorizations are individual in each case (but each factor, as well known, has a multiple of (p+1) sitting in it, so this "reduced variety" reverberates through the process and helps lcm to be a multiple of (p-1) for one reason or another.

Periods may (?) be different if we use the other popular seed values S0 = 10, or S0 = 2/3 (mod Mp).


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