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From: Bill Dubuque on 29 Jun 2010 21:01
Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote:
> Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> writes:
>> Pubkeybreaker <pubkeybreaker(a)aol.com> wrote:
>>>
>>> On Jun 29, 9:32 am, recoder <kurtulmeh...(a)gmail.com> wrote:
>>>> what is 2^(p-2) mod (p^2)
>>>> and what is the name of the topic in Number theory which handles such
>>>> congruences?
>>>
>>> 2^(p-2) mod p is just 2^-1 mod p == (p+1)/2
>>> Now apply Hensel's Lemma to lift to p^2.
>>
>> Hensel's Lemma?
>>
>> 2^(p-1) = 1 (mod p) (provided that p is prime)
>> but 2^(p-1) (mod p^2) is another kettle of fish.
>> Very little is known about it.
>> Only 2 primes are known for which 2^(p-1) = 1 (mod p^2).
>> The same would hold, mutatis mutandis, for 2^(p-2).
>
> Google "Wieferich Prime"

and "Fermat quotient".
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