60e to 70e done. Three new near Wilson primes found after the first one I already reported:
[CODE]64643245189 121p 66966581777 1+91p 67133912011 1+9p[/CODE] Reserving 250e9 to 270e9. 
[QUOTE=rajula;271945]60e to 70e done. Three new near Wilson primes found after the first one I already reported:[/QUOTE]
You picked a good range, so far all dry up here at this end. 
I'm now reserving:
[B]270e9 to 300e9[/B] And new discovery [CODE]231939720421 1+41p[/CODE] 
[QUOTE=Jeff Gilchrist;272134]I'm now reserving:
[B]270e9 to 300e9[/B] And new discovery [CODE]231939720421 1+41p[/CODE][/QUOTE] Please reduce my ignorance by explaining the notation. 
Now that we are getting up into higher numbers the memory usage is going up a lot. Trying to do 3e9 range with 45M primes is pushing over 4GB a process now. In order to run 2 processes at the same time I'm going to have to start cutting down to smaller ranges again.

[QUOTE=R.D. Silverman;272144]Please reduce my ignorance by explaining the notation.[/QUOTE]
(p1)! == 1+k*p mod p^2 for abs(k) <= 100 p=231939720421 k=41 
I completed the range 160e9200e9, found no near Wilson prime.
Reserving the range 300e9340e9. 
[QUOTE=R.D. Silverman;272144]Please reduce my ignorance by explaining the notation.[/QUOTE]
You probably have pieced it together already, but the idea is to quantify the 'closeness' to a Wilson prime by expressing the factorial in base p. The smaller the p multiple the closer to being 1 mod p^2. 
[QUOTE=jasonp;272403]You probably have pieced it together already, but the idea is to quantify the 'closeness' to a Wilson prime by expressing the factorial in base p. The smaller the p multiple the closer to being 1 mod p^2.[/QUOTE]
Note that 'closeness' as defined here conflicts with the notion of closeness over the padics. 
[QUOTE=R.D. Silverman;272408]Note that 'closeness' as defined here conflicts with the notion of closeness over the padics.[/QUOTE]
Do you mean that jason used the term "closeness" incorrectly? Could you elaborate on your statement? I don't understand what you mean when you say "closeness over the padics". I presume that is something different than what is referred to here as "special instance". 
[QUOTE=rogue;272415]Do you mean that jason used the term "closeness" incorrectly?
Could you elaborate on your statement? I don't understand what you mean when you say "closeness over the padics". I presume that is something different than what is referred to here as "special instance".[/QUOTE] Serre's "A Course In Arithmetic" contains a good exposition of padic fields. I can provide other references as well. I have neither the time (or the space!) in this forum to give a "Padics 101" course. 
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