Smooth as a newborn's rump: A P-1 Story

[NBtarheel_33]

My 6th P-1 success in 101 tries on V5, on M51197221. The factor (caught in Stage 1, of course) was about as small as you could/should get with P-1: 848763226512598523671, a mere 69.52 bits. The k-value is remarkably smooth:

k = 8289153297135 = 3^2 * 5 * 19 * 29 * 263 * 577 * 2203.

I wonder if TF'ing to 2^70 would have actually caught this one before P-1? Any thoughts?

[Mini-Geek]

I'm pretty sure these are, in order from best to worst, how you could've found the factor:
For me here are some rough timings, (for my CPU) as estimated by Prime95: (arranged from best to worst)
P-1 with B1=B2=2203: 10 minutes (by Status) or 12 minutes real-time.
P-1 with B1=577 and B2=2203: 3 minutes (by Status) or 10 minutes (what it really took me, with multithreading for stage 1/2 work and no multithreading on the GCD) or 13 minutes (my guesstimate for how long it'd really take with single-threaded work) (takes 298 MB to run all 8 relative primes at once)
TF from 2^69 to 2^70: 53 hours (if the full amount ran, which it wouldn't)
P-1 with 400 MB: about 53 hours (Prime95 reports 106 for both stages)
P-1 with 700 MB: about 56 hours (Prime95 reports 113 for both stages)

I'm not sure exactly how long it'd take for TF to find the factor. I know that Prime95 doesn't search the numbers from lowest to highest, and I think the potential factor's value mod 120(?) is involved, but I don't really know how to tell where in the 53 hour search it'd turn up. Either way, it'd find it faster than P-1 (with normal bounds) would. In case I'm right about the mod 120 thing, 2kp+1 = 31 mod 120, k = 15 mod 120. (not that I know how to interpret that, just putting it here in case someone knows )

In brief: Yes, you could've found it much faster by TF, or by better-selected P-1 bounds. But without knowing in advance what the factor was, P-1 as you ran was likely the most efficient option overall.

[Prime95]

Assuming Mini-Geek's numbers are right, a full P-1 run of 106 hours gave you an ~5% chance of finding a factor. A TF run of 53 hours would give you ~1.4% chance of finding a factor.

Yet, if you *knew* a factor was in the next bit-level, then TF will find it much faster.

[NBtarheel_33]

One of the birthday exponents cracked this time, again in Stage 1:

M47293063 has the factor 2037441373124298094871.

Again, a small fry, only 70.79 bits. Not quite as smooth as the last one, but still pretty darn smooth:

k = 21540594369245 = 5 x 7 x 2477 x 5209 x 47699.

That's 7 successes in 109 tries, or 6.42%. I seem to be on a bit of a P-1 roll .