fond of a factor? Urn yourself to become remains

[BudgieJane]

OK. I'm sorry for any confusion.

Consider M63,697,411. This can be expressed as a^n - b^n, where a = 2, b = 1 and n = 63697411.
Factor 1298457663977336673423680303 = 2kn + 1
n = 63697411
k = 19^2 × 29 × 37 × 761 × 94219 × 366983

Now consider what I'm asking, which is
If 2kn + 1 | a^n ± b^n and we know k and n, can we find a and b and, if so, how?
For an example of this I took your factor, and split it differently into k and n, giving
Number = a^n ± b^n
Factor 1298457663977336673423680303 = 2kn + 1, for different k and n
n = 366983
k = 19^2 × 29 × 37 × 761 × 94219 × 63697411

According to Legendre's theorem, this ought to be possible. I'm asking how we can do it.

Put it another way:
You started with a (=2), b (=1) and n (=63697411), and found factor p (=1298457663977336673423680303).
I'm starting with p (=1298457663977336673423680303) and n (=366983, which we know is a factor of p-1) and want to find a and b.
Can this be done, and, if so, how?

[Batalov]

For a=2, b=1, and unknown n (i.e. "We know a factor p of Mn, but lost the n value" with presumably prime n):

# pari/gp
? p=1298457663977336673423680303;
? f=factor((p-1)/2)[,1]
%4 = [19, 29, 37, 761, 94219, 366983, 63697411]~
? for(k=1,#f,if(Mod(2,p)^f[k]==1,print(f[k])))
63697411

If n is composite, use f=divisors(p-1).

For known n, unknown (a,b), with |b|<a<N: similar, but make an array of modular values and then scan pairs (a,b) to match them to be equal (or to add up to 0, for the aⁿ+bⁿ case).

For unknown n, unknown (a,b): combine these recipes.

[blahpy]

Originally Posted by blahpy

Currently 52% of the way through 94.3M to 94.4M from 2^65 to 2^66.

Done! 33 factors found in 2285 exponents.

[lycorn]

Did you eventually manage to report them? After the server hiccup of last morning, I mean.

[Jwb52z]

P-1 found a factor in stage #1, B1=580000.
UID: Jwb52z/Clay, M65699873 has a factor: 33015871761096951139589759831

94.737 bits.

[blahpy]

Originally Posted by lycorn

Did you eventually manage to report them? After the server hiccup of last morning, I mean.

Yeah, I sent them when I woke up this morning (it was last night here). All's fine now.

[TheJudger]

YACS2F (yet another composite stage #2 factor):

P-1 found a factor in stage #2, B1=580000, B2=11455000.
M65169077 has a factor: 819077031502508920172537251383103290199382793839 (159.16 Bits)

f₁ = 185525527686121837198207 (77.30 Bits)
f₂ = 4414902044576044881941777 (81.87 Bits)
k₁ = 3² * 149 * 174703 * 6075793
k₂ = 2³ * 83 * 1553 * 5659 * 5804573

New personal highscore for a prime P-1 factor:

P-1 found a factor in stage #2, B1=560000, B2=11200000, E=12.
M62720027 has a factor: 122667678181858045951591262815331625751850407 (146.45 Bits)
k = 3 * 13 * 31 * 83 * 107 * 3217 * 5099 * 43577 * 63377 * 167641 * 11992243

Oliver

[prgamma10]

Originally Posted by TheJudger

New personal highscore for a prime P-1 factor:

P-1 found a factor in stage #2, B1=560000, B2=11200000, E=12.
M62720027 has a factor: 122667678181858045951591262815331625751850407 (146.45 Bits)
k = 3 * 13 * 31 * 83 * 107 * 3217 * 5099 * 43577 * 63377 * 167641 * 11992243

Oliver

A 45-digit prime factor found with those bounds? Amazing!

[flashjh]

Originally Posted by TheJudger

...

New personal highscore for a prime P-1 factor:

P-1 found a factor in stage #2, B1=560000, B2=11200000, E=12.
M62720027 has a factor: 122667678181858045951591262815331625751850407 (146.45 Bits)
k = 3 * 13 * 31 * 83 * 107 * 3217 * 5099 * 43577 * 63377 * 167641 * 11992243

Oliver

It was also found with Brent-Suyama extension, but just barely. Which would partially explain why it is such a large prime factor. Quite an amazing find! When Oliver loads his data up to James' website it will show up on this link.

[Jwb52z]

P-1 found a factor in stage #1, B1=755000.
UID: Jwb52z/Clay, M69551917 has a factor: 29928764437580757262509857

84.630 bits.

[Prime95]

Originally Posted by TheJudger

New personal highscore for a prime P-1 factor:

P-1 found a factor in stage #2, B1=560000, B2=11200000, E=12.
M62720027 has a factor: 122667678181858045951591262815331625751850407 (146.45 Bits)
k = 3 * 13 * 31 * 83 * 107 * 3217 * 5099 * 43577 * 63377 * 167641 * 11992243

Is that a GIMPS P-1 record?