I came across 2 problematic numbers.

All below runs are on Core 2 Duo E6600 with Windows Vista Home Premium and
PFGW Version 1.2.0 for Windows [FFT v23.8]
I got similar results in undocumented runs with
PFGW Version 20041023.Win_Dev (Beta 'caveat utilitor') [FFT v23.8]

C:>pfgw problem.txt
982451653*2^42940-20987 is 3-PRP! (8.0767s+0.0003s)
1844504805*2^44100+569 is 3-PRP! (7.8038s+0.0016s)

C:>pfgw -a1 problem.txt
982451653*2^42940-20987 is composite: [FFF9A36E464F30D] (9.7057s+0.0002s)
1844504805*2^44100+569 is composite: [180D3A69890AB64] (9.1550s+0.0005s)

C:>pfgw -a2 problem.txt
982451653*2^42940-20987 is composite: [163249F1BDA528DD] (11.6043s+0.0002s)
1844504805*2^44100+569 is composite: [2AD9952EA23058D9] (11.1462s+0.0005s)

Other tested Fermat bases gave the same result:
-a1 and -a2 say composite with different residues. No -a argument says PRP.
The larger number is tested faster each time.

C:>pfgw -tc problem.txt
Primality testing 982451653*2^42940-20987 [N-1/N+1,
Brillhart-Lehmer-Selfridge]
Running N-1 test using base 2
982451653*2^42940-20987 is composite (12.1520s+0.0003s)
Primality testing 1844504805*2^44100+569 [N-1/N+1,
Brillhart-Lehmer-Selfridge]
Running N-1 test using base 3
1844504805*2^44100+569 is composite (12.2952s+0.0007s)

I usually consider -a1 and -a2 and -tc more reliable than no argument,
but here it turned out to be opposite.

From pfgwdoc.txt:
If -rm is used, then the modular reduction code is "tested". This is
very slow, but it can feret out problems in the reducer. This should
usually ONLY be used during developement, and trying to determine if
"problem" numbers are having problems with the FFT's or the reduction.

-rm was very slow indeed but still said PRP without an -a argument:

C:>pfgw -rm problem.txt
982451653*2^42940-20987 is 3-PRP! (175.8486s+0.0003s)
1844504805*2^44100+569 is 3-PRP! (187.6790s+0.0012s)

-rm with -a1 or -a2 immediately failed:

C:>pfgw -a1 -rm problem.txt
Error encountered in 'asm-Proth-like' modular reduction (Bit #14).
982451653*2^42940-20987 ERROR DURING PROCESSING! (0.0373s+0.0002s)
Error encountered in 'asm-Proth-like' modular reduction (Bit #14).
1844504805*2^44100+569 ERROR DURING PROCESSING! (0.0365s+0.0004s)

C:>pfgw -a2 -rm problem.txt
Error encountered in 'asm-Proth-like' modular reduction (Bit #14).
982451653*2^42940-20987 ERROR DURING PROCESSING! (0.0408s+0.0002s)
Error encountered in 'asm-Proth-like' modular reduction (Bit #14).
1844504805*2^44100+569 ERROR DURING PROCESSING! (0.0400s+0.0004s)

The numbers are prp according to a Fermat 3-PRP test and
5 Miller-Rabin tests by the GMP library.
PARI/GP's ispseudoprime makes a BPSW test and also says prp.
I'm convinced the numbers are prp, but their status is not important.

Is "'asm-Proth-like' modular reduction" only used on k*2^n+-c for c>1?
Is it known whether -a1 and -a2 are normally less reliable than
no -a for that form?

--
Jens Kruse Andersen