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?

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):
[CODE]# 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
[/CODE]
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 a[SUP]n[/SUP]+b[SUP]n[/SUP] case).

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

[QUOTE=blahpy;347548]Currently 52% of the way through 94.3M to 94.4M from 2^65 to 2^66.[/QUOTE]

Done! 33 factors found in 2285 exponents.

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

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

94.737 bits.

[QUOTE=lycorn;348067]Did you eventually manage to report them? After the server hiccup of last morning, I mean.[/QUOTE]

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

YACS2F ([B]y[/B]et [B]a[/B]nother [B]c[/B]omposite [B]s[/B]tage #[B]2[/B] [B]f[/B]actor):[INDENT]P-1 found a factor in stage #2, B1=580000, B2=11455000.
M65169077 has a factor: 819077031502508920172537251383103290199382793839 (159.16 Bits)

f[SUB]1[/SUB] = 185525527686121837198207 (77.30 Bits)
f[SUB]2[/SUB] = 4414902044576044881941777 (81.87 Bits)
k[SUB]1[/SUB] = 3[SUP]2[/SUP] * 149 * 174703 * 6075793
k[SUB]2[/SUB] = 2[SUP]3[/SUP] * 83 * 1553 * 5659 * 5804573[/INDENT]

New personal [B]highscore[/B] for a [B]prime P-1 factor[/B]:[INDENT]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
[/INDENT]Oliver

[QUOTE=TheJudger;348198]
New personal [B]highscore[/B] for a [B]prime P-1 factor[/B]:[INDENT]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
[/INDENT]Oliver[/QUOTE]
A 45-digit prime factor found with those bounds? Amazing!

[QUOTE=TheJudger;348198]...

New personal [B]highscore[/B] for a [B]prime P-1 factor[/B]:[INDENT]P-1 found a factor in stage #2, B1=560000, B2=[B]11200000[/B], E=12.
M[URL="http://www.mersenne.ca/exponent.php?exponentdetails=62720027"]62720027[/URL] has a factor: 122667678181858045951591262815331625751850407 (146.45 Bits)
k = 3 * 13 * 31 * 83 * 107 * 3217 * 5099 * 43577 * 63377 * 167641 * [B]11992243[/B]
[/INDENT]Oliver[/QUOTE]It was also found with [I][URL="http://www.mersennewiki.org/index.php/Brent-Suyama_extension"]Brent-Suyama extension[/URL], [/I]but just barely. Which would partially explain why it is such a large prime factor. [B]Quite an amazing find![/B] When Oliver loads his data up to James' website it will show up on this [URL="http://www.mersenne.ca/brent-suyama.php"]link[/URL].

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

84.630 bits.

[QUOTE=TheJudger;348198]
New personal [B]highscore[/B] for a [B]prime P-1 factor[/B]:[INDENT]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
[/INDENT][/QUOTE]

Is that a GIMPS P-1 record?